FINITE ELEMENT ANALYSIS OF STRESS RUPTURE IN …

FINITE ELEMENT ANALYSIS OF STRESS RUPTURE IN PRESSURE VESSELS EXPOSED TO ACCIDENTAL FIRE

LOADING

by

CHRISTOPHER CORNELIU MANU

A thesis submitted to the Department of Mechanical and Materials Engineering

In conformity with the requirements for

the degree of Master of Science (Engineering)

Queen’s University

Kingston, Ontario, Canada

(July, 2008)

Copyright © Christopher Corneliu Manu, 2008

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Abstract

A numerical model that predicts high temperature pressure vessel rupture was developed.

The finite element method of analysis was used to determine the effects that various parameters

had on pressure vessel failure. The work was concerned with 500, 1000 and 33000 US gallon

pressure vessels made of SA 455 steel. Experimental pressure vessel fire tests have shown that

vessel rupture in a fully engulfing fire can occur in less than 30 minutes. This experimental work

was used both to validate the numerical results as well as to provide important vessel temperature

distribution information.

Due to the fact that SA 455 steel is not meant for high temperature applications, there

was little published high temperature material data. Therefore, elevated temperature tensile tests

and creep rupture tests were performed to measure needed material properties. Creep and creep

damage constants were calculated from SA 455 steel’s creep rupture data.

The Kachanov One-State Variable technique and the MPC Omega method were the creep

damage techniques chosen to predict SA 455 steel’s high temperature time-dependent behaviour.

The specimens used in the mechanical testing were modeled to numerically predict the creep

rupture behaviour measured in the lab. An extensive comparison between the experimental and

numerical uniaxial creep rupture results revealed that both techniques could adequately predict

failure times at all tested conditions; however, the MPC Omega method was generally more

accurate at predicting creep failure strains. The comparison also showed that the MPC Omega

method was more numerically stable than the One-State Variable technique when analyzing SA

455 steel’s creep rupture.

The creep models were modified to account for multiaxial states of stress and were used

to analyze the high temperature failure of pressure vessels. The various parameters considered

included pressure vessel dimensions, fire type (fully engulfing or local impingement), peak wall

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temperature and internal pressure. The objective of these analyses was to gain a better

understanding of the structural failure of pressure vessels exposed to various accidental fire

conditions.

The numerical results of rupture time and geometry of failure region were shown to agree

with experimental fire tests. From the fully engulfing fire numerical analyses, it was shown that

pressure vessels with a smaller length to diameter ratio and a larger thickness to diameter ratio

were inherently safer. It was also shown that as the heated area was reduced, the failure time

increased for the same internal pressure and peak wall temperature. Therefore, fully engulfing

fires produced more structurally unstable conditions in pressure vessels then local fire

impingements.

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Acknowledgements

My thanks go to Dr. A.M. Birk and Dr. I.Y. Kim of the Department of Mechanical &

Materials Engineering of Queen’s University for the guidance and support they provided me with

during this project.

This project has been funded by contributions from the Natural Sciences and Engineering

Research Council of Canada (NSERC) and the author was also assisted by a Queen’s Graduate

Award.

Further thanks are extended to Dr. B.J. Diak for the input and guidance he patiently

provided during the high temperature mechanical testing of SA 455 steel. Furthermore, I would

like to thank all the technical staff in McLaughlin and Nicol Hall for their support and assistance

in development of the experimental test rig.

I would like to thank my family, friends and colleagues for their patience, advice and

assistance throughout the duration of this project.

I would also like to thank my significant other, Katherine, for her support during these 2

years, without her pushing me to get up in the morning this might have taken 3 years.

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Table of Contents Abstract ............................................................................................................................................ii Acknowledgements........................................................................................................................iiv Table of Contents............................................................................................................................. v List of Figures ...............................................................................................................................viii List of Tables ..............................................................................................................................xviii

Nomenclature............................................................................................................................xviiix

Chapter 1 Introduction ................................................................................................................... 1

1.1 Motivation..................................................................................................................... 1

1.2 Pressure Vessel Failure ................................................................................................ 1

1.3 Objectives .................................................................................................................... 4

Chapter 2 Literature Review.......................................................................................................... 7

2.1 Mechanical Testing...................................................................................................... 7

2.1.1 Tension Tests of Metallic Materials............................................................... 7

2.1.2 Elevated Temperature Tension Tests of Metallic Materials .......................... 8

2.1.3 Creep Rupture Tests of Metallic Materials .................................................... 8

2.2 Numerical Creep Failure Prediction ............................................................................ 9

2.2.1 Creep ........................................................................................................... 10

2.2.2 Kachanov/Rabotnov (Empirical based Continuum Damage Mechanics) .... 16

2.2.3 MPC Omega Method ................................................................................... 17

2.2.4 Theta Projection Concept............................................................................. 20

2.2.5 Physically Based Continuum Damage Mechanics....................................... 23

2.2.6 Robinson's Life Fraction Rule...................................................................... 26

2.2.7 Larson-Miller ............................................................................................... 27

2.3 Pressure Vessel Failure Analysis ............................................................................... 29

2.3.1 Thermal Response of a Pressure Vessel in an Accidental Fire .................... 30

2.3.2 Pressure Vessel Failure due to Accidental Fire Loading ............................. 33

2.3.3 Computer Modeling of Pressure Vessel Failure in Accidental Fire

Loadings....................................................................................................... 35

Chapter 3 Theory ......................................................................................................................... 41

3.1 High Temperature Steel Failure................................................................................. 41

3.1.1 Plasticity....................................................................................................... 42

3.1.2 Creep ............................................................................................................ 47

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3.1.3 Creep Damage Equations............................................................................. 54

3.1.3.1 Kachanov One-State Variable Technique.............................................. 55

3.1.3.2 MPC Omega Method ............................................................................. 56

3.2 FEA Governing Equations ......................................................................................... 58

3.3 FEA User Subroutines ............................................................................................... 59

Chapter 4 SA 455 Mechanical Testing ........................................................................................ 62

4.1 Experimental Specimens............................................................................................ 62

4.2 Experimental Apparatus............................................................................................. 63

4.3 Data Acquisition ........................................................................................................ 67

4.4 Experimental Details.................................................................................................. 69

4.5 Elevated Temperature Tensile Tests .......................................................................... 71

4.6 Creep Rupture Tests................................................................................................... 74

4.7 Creep Data Correlation .............................................................................................. 78

Chapter 5 SA 455 Creep and Creep Damage Constants .............................................................. 83

5.1 Kachanov One-State Variable Constants ................................................................... 83

5.2 MPC Omega Constants.............................................................................................. 86

Chapter 6 Numerical Analysis of Tensile Specimens.................................................................. 90

6.1 Tensile Specimen Model............................................................................................ 90

6.1.1 Analysis Geometry and Mesh ...................................................................... 91

6.1.2 Kachanov One-State Variable...................................................................... 96

6.1.3 MPC Omega................................................................................................. 96

6.2 Results and Discussion .............................................................................................. 97

Chapter 7 Numerical Pressure Vessel Analysis......................................................................... 107

7.1 Geometry and Mesh................................................................................................. 109

7.1.1 500 US gallon Pressure Vessel .................................................................. 111

7.1.2 1000 US gallon Pressure Vessel ................................................................ 114

7.1.3 33000 US gallon Pressure Vessel .............................................................. 116

7.2 Creep Damage Models............................................................................................. 118

7.2.1 Kachanov One-State Variable Analyses .................................................... 118

7.2.2 MPC Omega Analyses ............................................................................... 119

7.3 Loading Conditions.................................................................................................. 119

7.4 Time Independent Temperature and Pressure Distribution Loading ....................... 120

7.4.1 Description of Time Independent Temperature and Pressure Distribution

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Loading ...................................................................................................... 120

7.4.2 Results and Discussion of Time Independent Temperature and Pressure

Distribution Loading .................................................................................. 126

7.4.2.1 500 US gallon Pressure Vessel ................................................. 126

7.4.2.2 1000 US gallon Pressure Vessel ............................................... 135

7.4.2.3 33000 US gallon Pressure Vessel ............................................ 138

7.5 Time Independent Hot Spots.................................................................................... 141

7.5.1 Description of Time Independent Hot Spot Loading................................. 142

7.5.2 Results and Discussion of Time Independent Hot Spot Loading............... 145

Chapter 8 Conclusions and Recommendations.......................................................................... 155 8.1 Conclusions.............................................................................................................. 155 8.1.1 Creep Damage Models............................................................................... 155

8.1.2 Commercial Finite Element Analysis computer Codes ............................. 156 8.1.3 Fully Engulfing Fire................................................................................... 156 8.1.4 Local Fire Impingement (Hot Spots) ......................................................... 157 8.2 Recommendations.................................................................................................... 158

References.................................................................................................................................... 159 Appendix A - SA 455 Steel Measured Creep Curves ................................................................. 164 Appendix B - Detailed Drawings of 500, 1000 and 33000 US gallon Pressure Vessels ............. 168 Appendix C - Results of Numerical Analysis for Time-Independent Temperature and Pressure

Loadings................................................................................................................ 172 Appendix D - Results of Numerical Analysis for Time-Independent Hot Spot Loadings .......... 181

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List of Figures Figure 1-1: Experimental pressure vessel fire test (Birk private communication, 2008) ................ 2

Figure 2-1: Creep failure time dependence on temperature and stress (Kraus 1980) ................... 12

Figure 2-2: Creep strain vs. time; typical plot generated from a creep rupture test (Kraus, 1980)

....................................................................................................................................................... 13

Figure 2-3: Temperature dependence of a typical creep curve (Kraus, 1980) .............................. 14 Figure 2-4: Stress dependence of a typical creep curve (Kraus, 1980) ......................................... 14 Figure 2-5: Effect of stress level on relaxation at constant temperature (Kraus, 1980) ................ 15 Figure 2-6: (a) Tested and fitted uniaxial creep strain curves of the Bar 257 steel at 650 oC for the

one-state variable equation. (b) Tested and fitted uniaxial creep strain curves of

the Bar 257 steel at 650 oC for the three-state variable equation (Hyde et al.,

2006) ................................................................................................................... 17 Figure 2-7: Deriving Ωp from experimental data (Prager, 1994) ................................................... 19 Figure 2-8: Theta projection modelling of +5oC thermal case, von-Mises stresses (Law et al.,

2002) ........................................................................................................................ 22 Figure 2-9: Variation of creep rate at inner and outer walls for +5oC case (Law et al., 2002) ..... 23 Figure 2-10: Typical Larson-Miller plot........................................................................................ 28 Figure 2-11: Thermally protected pressure vessel shell temperatures (Droste and Schoen, 1988)

....................................................................................................................................................... 30 Figure 2-12: Thermally unprotected peak vessel wall temperatures; various initial fill levels

(Moodie et al., 1988)................................................................................................ 31 Figure 2-13: Wall thermocouple layout for test 04-3 (Birk et al., 2006) ...................................... 32

Figure 2-14: Thermally protected pressure vessel wall temperature, test 04-3 (15% area insulation

defect); wall 12, 13 and 14 are on outer surface on thermal protection (Birk et al.,

2006) ...................................................................................................................... 33

Figure 2-15: Comparison of calculated tank bursting pressures and test results (Droste and

Schoen, 1988) ....................................................................................................... 34 Figure 2-16: Predicted and measured tank wall temperatures vs time from fire ignition for

upright uninsulated fifth-scale tank-car exposed to an engulfing fire (Birk, 1988)

....................................................................................................................................................... 37 Figure 2-17: Predicted tank wall stresses and material strength vs time from fire ignition for

upright uninsulated fifth-scale tank-car exposed to an engulfing fire (Birk, 1988)37 Figure 2-18: Predicted tank wall stresses and material strength vs time from fire ignition for

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upright uninsulated full-scale tank-car exposed to an engulfing fire (Birk, 1988) 38 Figure 3-1: 0.2% offset used to define tensile yield strength (Hibbeler, 2003) ............................ 43

Figure 3-2: Loading path generated by the isotropic hardening rule (ANSYS, 2004) ................. 46 Figure 3-3: Loading path generated by the kinematic hardening rule (ANSYS, 2004) ................ 47

Figure 3-4: Creep strain vs. time; typical plot generated from a creep rupture test (Kraus, 1980)

....................................................................................................................................................... 48 Figure 3-5: Creep strain history prediction from time hardening theory (Kraus, 1980) .............. 50

Figure 3-6: Creep strain history prediction from strain hardening theory (Kraus, 1980) ............. 51 Figure 3-7: Where various user subroutines fit into FEA (ABAQUS, 2007) ............................... 61 Figure 4-1: Specimen dimensions, the reduced area in the centre is the gauge length; specimen

thickness is 7.1, all dimensions are in mm .............................................................. 63 Figure 4-2: Instron with furnace, custom grips and sample........................................................... 64 Figure 4-3: Custom grip’s design drawing .................................................................................... 65 Figure 4-4: Custom tensile test grip............................................................................................... 65 Figure 4-5: Custom clamshell furnace ........................................................................................... 66 Figure 4-6: Omega CN2100-R20 temperature controller .............................................................. 66 Figure 4-7: Data acquisition system used to collect sample temperatures along the gage length .68

Figure 4-8: Temperature history of top and bottom of gauge length during a creep test at 630oC;

start up heating, 20 minute hold time and creep test time are shown ...................... 68

Figure 4-9: Temperature history of top and bottom of gauge length during a creep test at 630oC;

20 minute hold time and 5.7 minute test are shown................................................ 69 Figure 4-10: Failed specimen with 3 thermocouples spot welded onto it ..................................... 71

Figure 4-11: Comparison of ultimate tensile strength of SA 455, TC 128 and medium carbon

steel. .............................................................................................................................................. 73

Figure 4-12: SA 455 steel true stress true strain plot at various temperatures............................... 74 Figure 4-13: Plot of SA 455 steel's stress rupture data .................................................................. 76 Figure 4-14: Plot of TC 128 steel’s stress rupture data (Birk et al., 2006) ................................... 76

Figure 4-15: Measured SA 455 steel creep curves at 600oC.......................................................... 77 Figure 4-16: Investigating the suitability of CL-M=20 for SA 455 steel ......................................... 79 Figure 4-17: Larson-Miller correlation for SA 455 steel, CL-M=19 ............................................... 80 Figure 4-18: Correlated vs measured failure times ........................................................................ 81 Figure 5-1: Relationship between natural logarithm of rupture time and natural logarithm of

initial stress; plot needed to compute χ and B’ for SA 455...................................... 84

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Figure 5-2: Temperature dependence of A’ and B’ ...................................................................... 86

Figure 5-3: Temperature dependence of n, φ , χ............................................................................ 86

Figure 5-4: Determining SA 455 steel’s value of Omega for a test temperature of 600oC and stress

of 177.3 MPa............................................................................................................ 87 Figure 5-5: Stress dependency of Ωp ............................................................................................. 88

Figure 5-6: Stress dependency of 0ε& .............................................................................................. 89

Figure 6-1: Numerical model of the tensile specimen’s gauge length........................................... 91 Figure 6-2: Numerical analysis boundary conditions and loads, bottom face is fixed in all

directions, pressure load applied on top face ........................................................... 92 Figure 6-3: ABAQUS meshed specimen gauge length ................................................................. 93 Figure 6-4: ANSYS meshed specimen gauge length..................................................................... 94 Figure 6-5: Mesh convergence using maximum von Mises stress ................................................ 95 Figure 6-6: Mesh convergence using maximum displacement...................................................... 95 Figure 6-7: Uniaxial creep validation of finite element analyses, T=550oC.................................. 98 Figure 6-8: Uniaxial creep validation of finite element analyses, T=600oC.................................. 99 Figure 6-9 Uniaxial creep validation of finite element analyses, T=630oC................................. 100 Figure 6-10 Uniaxial creep validation of finite element analyses, T=660oC ............................... 101 Figure 6-11: Uniaxial creep validation of finite element analyses, T=690oC.............................. 102 Figure 6-12: Uniaxial creep validation of finite element analyses, T=720oC.............................. 103 Figure 6-13: Predicted failure time .............................................................................................. 103 Figure 6-14: Predicted creep strain at failure............................................................................... 104

Figure 6-15: Typical ABAQUS creep damage contour plot at failure ........................................ 104 Figure 6-16: Typical ANSYS creep damage contour plot at failure............................................ 105 Figure 6-17: Creep damage accumulation using the Kachanov technique; T=600oC S=177.3

MPa, solved with Abaqus .................................................................................... 106

Figure 6-18: Creep damage accumulation using the MPC Omega method; T=600oC S=177.3

MPa, solved with Abaqus .................................................................................... 106 Figure 7- 1: 500 US gallon pressure vessel model....................................................................... 111 Figure 7-2: 500 US gallon pressure vessel model with pressure loads shown as arrows ............ 112 Figure 7-3: 500 US gallon pressure vessel model, view of underside; boundary conditions on the

left fixed all translational degrees of freedom and rotation about the 2 and 3

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directions, boundary condition on the right fixed translation in the 1 and 2 directions

..................................................................................................................................................... 112 Figure 7-4: 500 US gallon pressure vessel mesh ......................................................................... 113 Figure 7-5: 500 US gallon pressure vessel mesh, zoom in view of end detail............................. 113 Figure 7- 6: 1000 US gallon pressure vessel model..................................................................... 114 Figure 7-7: 1000 US gallon pressure vessel model, view of underside; boundary conditions on the

left fixed all translational degrees of freedom and rotation about the 2 and 3

directions, boundary condition on the right fixed translation in the 1 and 2 directions

..................................................................................................................................................... 115 Figure 7-8: 1000 US gallon pressure vessel mesh ....................................................................... 115 Figure 7- 9: 33000 US gallon pressure vessel model................................................................... 116 Figure 7-10: 33000 US gallon pressure vessel model, view of underside; boundary conditions on

the left fixed all translational degrees of freedom and rotation about the 2 and 3

directions, boundary condition on the right fixed translation in the 1 and 2

directions.............................................................................................................. 117 Figure 7-11: 1000 US gallon pressure vessel mesh ..................................................................... 117 Figure 7-12: Cross section representation of pressure vessel temperature distribution showing

angle measurement convention............................................................................ 122 Figure 7-13: Sensitivity study of size of frothing region ............................................................. 123 Figure 7-14: Sensitivity study of value of thermal gradient in the vapour wall space................. 124 Figure 7-15: Piecewise temperature distribution; theta is measured from top centre of vessel.

δ=0.1, β=20o, TH=650oC, TC=130oC, fill =50% or 90o ...................................... 125 Figure 7-16: Imposed vessel temperature distribution (oC); 500 US gallon pressure vessel. δ=0.1,

β=20o, TH=650oC, TC=130oC, fill =50% or 90o ................................................. 126 Figure 7-17: 500 US gallon pressure vessel predicted failure times (MPC Omega, ABAQUS) 127 Figure 7-18: Predicted MPC Omega creep damage at time of failure, T=650oC p=2.07MPa (300

psig); solved with ABAQUS................................................................................ 129 Figure 7-19: Predicted MPC Omega effective creep strain at time of failure, T=650oC p=2.07MPa

(300 psig); solved with ABAQUS ....................................................................... 129 Figure 7-20: Predicted MPC Omega effective plastic strain at time of failure, T=650oC

p=2.07MPa (300 psig); solved with ABAQUS ................................................... 130 Figure 7-21: Time history of MPC Omega effective creep strain at point of eventual failure,

T=650oC p=2.07MPa (300 psig); solved with ABAQUS.................................... 131

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Figure 7-22: Time history of effective stress at point of eventual failure, T=650oC p=2.07MPa

(300 psig); solved with ABAQUS ....................................................................... 131 Figure 7-23: Failed fire tested 500 US gallon pressure vessel; 25% engulfing fire, peak wall

temperature = 740oC, peak internal pressure = 2.59MPa (375 psig) ................... 132 Figure 7-24: Predicted One-State Variable Technique creep damage at time of failure, T=650oC

p=2.07MPa (300 psig); solved with ABAQUS ................................................... 134 Figure 7-25: Predicted One-State Variable Technique effective creep strain at time of failure,

T=650oC p=2.07MPa (300 psig); solved with ABAQUS.................................... 134 Figure 7-26: Predicted One-State Variable Technique effective plastic strain at time of failure,

T=650oC p=2.07MPa (300 psig); solved with ABAQUS.................................... 135 Figure 7-27: 1000 US gallon vessel predicted MPC Omega creep damage at time of failure,

T=650oC p=2.07MPa (300 psig); solved with ABAQUS.................................... 136 Figure 7-28: 1000 US gallon vessel predicted MPC Omega effective creep strain at time of

failure, T=650oC p=2.07MPa (300 psig); solved with ABAQUS....................... 137 Figure 7-29: 1000 US gallon vessel predicted MPC Omega effective plastic strain at time of

failure, T=650oC p=2.07MPa (300 psig); solved with ABAQUS....................... 137 Figure 7-30: 33000 US gallon vessel predicted MPC Omega creep damage at time of failure,

T=620oC p=2.07MPa (300 psig); solved with ABAQUS.................................... 139 Figure 7-31: 33000 US gallon vessel predicted MPC Omega effective creep strain at time of

failure, T=620oC p=2.07MPa (300 psig); solved with ABAQUS....................... 140 Figure 7-32: 33000 US gallon vessel predicted MPC Omega effective plastic strain at time of

failure, T=620oC p=2.07MPa (300 psig); solved with ABAQUS....................... 140 Figure 7-33: 500 US gallon pressure vessel part model for hot spot analysis (100mm radius)... 143 Figure 7-34: 500 US gallon pressure vessel mesh for hot spot analysis (100mm radius); 14,058

ABAQUS shell finite elements ............................................................................ 143 Figure 7-35: Detailed view of top dead centre of 500 US gallon pressure vessel mesh for hot spot

analysis (100mm radius) ...................................................................................... 144 Figure 7-36: Experimentally measured temperature distribution of a pressure vessel with a 25.4

cm defect (Birk and VanderSteen, 2003)............................................................. 145

Figure 7-37: Comparison between fully engulfing fire and hot spot failure times ...................... 146 Figure 7-38: 500 US gallon pressure vessel, imposed temperature distribution for hot spot

analysis (100mm radius); maximum temperature of 720oC, pressure of 2.07MPa

(300 psig), solved using ABAQUS...................................................................... 147

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Figure 7-39: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of

MPC Omega creep damage at time of failure; maximum temperature of 720oC,

pressure of 2.07MPa (300 psig), solved using ABAQUS.................................... 148 Figure 7-40: 500 US gallon pressure vessel hot spot analysis (100mm radius), time history of

MPC Omega creep damage at location of eventual failure; red plot is inner surface,

green plot is outer surface; maximum temperature of 720oC, pressure of 2.07MPa

(300 psig), solved using ABAQUS...................................................................... 148 Figure 7-41: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of

MPC Omega effective creep strain at time of failure; maximum temperature of

720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS........................ 149 Figure 7-42: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of

MPC Omega effective creep strain at time of failure; zoomed in on failure zone;

maximum temperature of 720oC, pressure of 2.07MPa (300 psig), solved using

ABAQUS ............................................................................................................. 150 Figure 7-43: 500 US gallon pressure vessel hot spot analysis (100mm radius), time history of

MPC Omega effective creep strain at location of eventual failure; red plot is inner

surface, green plot is outer surface; maximum temperature of 720oC, pressure of

2.07MPa (300 psig), solved using ABAQUS ...................................................... 150 Figure 7-44: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of

effective plastic strain at time of failure; maximum temperature of 720oC, pressure

of 2.07MPa (300 psig), solved using ABAQUS .................................................. 151 Figure 7-45: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of

hoop stress at time of failure; maximum temperature of 720oC, pressure of

2.07MPa (300 psig), solved using ABAQUS ...................................................... 152 Figure 7-46: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of

hoop stress at time of failure; zoomed in on failure region; maximum temperature

of 720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS ................... 152

Figure 7-47: 500 US Gallon pressure vessel hot spot analysis (100mm radius), time history of von

Mises stress at location of eventual failure; red plot is inner surface, green plot is

outer surface; maximum temperature of 720oC, pressure of 2.07MPa (300 psig),

solved using ABAQUS ........................................................................................ 153

Figure 7-48: Experimentally observed rupture of 500 US gallon pressure vessel with small

insulation defect (8% of tank surface); Peak wall temp=700oC, peak internal

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pressure = 2.59Mpa (375 psig) ............................................................................ 154

Figure A-1: SA 455 steel creep curve; T=550oC ......................................................................... 165 Figure A-2: SA 455 steel creep curve; T=600oC ......................................................................... 165 Figure A-3: SA 455 steel creep curve; T=630oC ......................................................................... 166 Figure A-4: SA 455 steel creep curve; T=660oC ......................................................................... 166 Figure A-5: SA 455 steel creep curve; T=690oC ......................................................................... 167 Figure A-6: SA 455 steel creep curve; T=720oC ......................................................................... 167 Figure B-1: Detailed engineering drawing of 500 US gallon pressure vessel ............................. 169 Figure B-2: Detailed engineering drawing of 1,000 US gallon pressure vessel .......................... 170 Figure B-3: Detailed engineering drawing of 33,000 US gallon pressure vessel ........................ 171 Figure C-1: Predicted MPC Omega creep damage at time of failure, T=620oC p=2.07MPa (300

psig); solved with ABAQUS.................................................................................. 173 Figure C-2: Predicted MPC Omega effective creep strain at time of failure, T=620oC p=2.07MPa

(300 psig); solved with ABAQUS ......................................................................... 173 Figure C-3: Predicted MPC Omega creep damage at time of failure, T=620oC p=2.24MPa (325

psig); solved ith ABAQUS .................................................................................... 174 Figure C-4: Predicted MPC Omega effective creep strain at time of failure, T=620oC p=2.24MPa


psig); solved with ABAQUS.................................................................................. 175 Figure C-6: Predicted MPC Omega effective creep strain at time of failure, T=650oC p=1.90MPa

(275 psig); solved with ABAQUS ......................................................................... 175 Figure C-7: Predicted MPC Omega creep damage at time of failure, T=650oC p=2.24MPa

(325 psig); solved with ABAQUS ......................................................................... 176 Figure C-8: Predicted MPC Omega effective creep strain at time of failure, T=650oC p=2.24MPa


psig); solved with ABAQUS.................................................................................. 177 Figure C-10: Predicted MPC Omega effective creep strain at time of failure, T=680oC

p=1.90MPa (275 psig); solved with ABAQUS ................................................... 177 Figure C-11: Predicted MPC Omega creep damage at time of failure, T=680oC p=2.07MPa

(300 psig); solved with ABAQUS ....................................................................... 178

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Figure C-12: Predicted MPC Omega effective creep strain at time of failure, T=680oC

p=2.07MPa (300 psig); solved with ABAQUS ................................................... 178 Figure C-13: Predicted MPC Omega creep damage at time of failure, T=680oC p=2.24MPa

(325 psig); solved with ABAQUS ....................................................................... 179 Figure C-14: Predicted MPC Omega effective creep strain at time of failure, T=680oC

p=2.24MPa (325 psig); solved with ABAQUS ................................................... 179 Figure C-15: Predicted effective plastic strain at time of failure, T=680oC p=2.24MPa

(325 psig); solved with ABAQUS ....................................................................... 180 Figure D-1: 500 US Gallon pressure vessel, imposed temperature distribution for a 100mm hot

spot analysis; maximum temperature of 680oC, pressure of 2.07MPa (300 psig),

solved using ABAQUS ......................................................................................... 182 Figure D-2: 500 US Gallon pressure vessel, contour plot of MPC Omega creep damage at time of

failure; 100mm radius hot spot, maximum temperature of 680oC, pressure of

2.07MPa (300 psig), solved using ABAQUS ........................................................ 182 Figure D-3: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at

time of failure; 100mm radius hot spot, maximum temperature of 680oC, pressure

of 2.07MPa (300 psig), solved using ABAQUS ................................................... 183 Figure D-4: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at


of 2.07MPa (300 psig), solved using ABAQUS; zoomed in view of failure region

..................................................................................................................................................... 183 Figure D-5: 500 US Gallon pressure vessel, contour plot of effective plastic strain at time of


2.07MPa (300 psig), solved using ABAQUS ........................................................ 184 Figure D-6: 500 US Gallon pressure vessel, imposed temperature distribution for a 100mm hot


solved using ABAQUS ......................................................................................... 184 Figure D-7: 500 US Gallon pressure vessel, contour plot of MPC Omega creep damage at time of


2.07MPa (300 psig), solved using ABAQUS ....................................................... 185 Figure D-8: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at


of 2.07MPa (300 psig), solved using ABAQUS ................................................... 185

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Figure D-9: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at





2.07MPa (300 psig), solved using ABAQUS ...................................................... 186 Figure D-11: 500 US Gallon pressure vessel, imposed temperature distribution for a 75mm hot


solved using ABAQUS ......................................................................................... 187 Figure D-12: 500 US Gallon pressure vessel, contour plot of MPC Omega creep damage at time

of failure; 75mm radius hot spot, maximum temperature of 700oC, pressure of

2.07MPa (300 psig), solved using ABAQUS ..................................................... 187 Figure D-13: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at


of 2.07MPa (300 psig), solved using ABAQUS .................................................. 188 Figure D-14: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at





2.07MPa (300 psig), solved using ABAQUS ..................................................... 189

Figure D-16: 500 US Gallon pressure vessel, imposed temperature distribution for a 75mm hot


solved using ABAQUS ......................................................................................... 189 Figure D-17: 500 US Gallon pressure vessel, contour plot of MPC Omega creep damage at time

of failure; 75mm radius hot spot, maximum temperature of 720oC, pressure of

2.07MPa (300 psig), solved using ABAQUS ..................................................... 190 Figure D-18: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at


of 2.07MPa (300 psig), solved using ABAQUS .................................................. 190 Figure D-19: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at

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2.07MPa (300 psig), solved using ABAQUS ..................................................... 191

xviii

List of Tables Table 2-1: Experimentally derived Omega values for carbon steel (Prager, 1994) ...................... 19 Table 2-2: Predicted failure times for plain and notched tubes with experimental data for

comparison. Test conditions: 565°C, 36.6 MPa. (Yang et al., 2004) ........................ 25 Table 2-3: Summary of Tank2004 stress rupture predictions compared to actual failure times.... 40 Table 4-1: Chemical composition of SA 455................................................................................. 62 Table 4-2: Tensile properties of SA 455........................................................................................ 63 Table 4-3: SA 455 temperature dependency of yield stress and UTS ........................................... 72 Table 4-4: Stress-rupture properties of SA 455 steel ..................................................................... 75 Table 4-5: Minimum creep rates of SA 455 steel .......................................................................... 78 Table 4-6: Larson-Miller parameter values ................................................................................... 80 Table 5-1: One-state variable continuum damage mechanics material constants for SA 455 ....... 85 Table 5-2: SA 455 steel Omega MPC creep constants .................................................................. 88 Table 7-1: MPC Omega numerically predicted failure times [min] of pressure vessels with respect

to internal pressure and maximum temperature solved with ABAQUS ................. 127 Table 7-2: Kachanov One-State Variable Technique numerically predicted failure times [min] of

pressure vessels with respect to internal pressure and maximum temperature solved

with ABAQUS ........................................................................................................ 133

Table 7-3: Comparison between 500 and 1000 US gallon analysis results; peak wall temp =

650oC, p=2.07MPa (300 psig)................................................................................. 138

Table 7-4: Comparison between 500 and 33000 US gallon analysis results; peak wall temp =

620oC, p=2.07MPa (300 psig)................................................................................. 141

Table 7-5: Summary of hot spot failure analysis of 500 US gallon pressure vessels .................. 145 Table 7-6: Relationship between hot spot radius and thermal defect radius................................ 146

xix

Nomenclature a,b,c,d Theta Projection Concept creep constants

A,n Creep constants shared by Norton and Kachanov

A’, χ, B’ One-State Variable equation creep constants

b Body force vector

C System damping matrix

CL-M Larson-Miller constant

d Symmetric component of gradient of the velocity vector

e Euler-Almansi strain tensor

e Base of natural logarithm

F Global force matrix

K Global stiffness matrix

m Creep constant shared by Norton, Kachanov and MPC Omega

M System mass matrix

PL-M Larson-Miller parameter

p, c MPC Omega creep constants

r Kachanov creep constant

s Surface in spatial description

S Engineering stress

Sij Deviatoric stress component

T Temperature

t Time

ts Time in service

tr Rupture time

t Traction vector

u Displacement vector

v Volume in spatial description

v Velocity vector

xx

Greek

α One-State Variable equation creep constant related to multiaxial stress-state

behaviour

αi, βi, γi, δi Theta Projection Concept creep constants

λd Plastic compliance

ε True strain

nε Engineering strain

ε Effective strain cε Uniaxial creep strain cε& Uniaxial creep strain rate .cε Effective creep strain rate cijε& Multiaxial creep strain rate component

0ε& MPC Omega creep constant, initial creep strain rate

pε True plastic strain

Ω Surface

pΩ MPC Omega creep constant, omega

σ True stress

nσ Engineering stress

mσ Hydrostatic stress

σ Effective stress

rσ One-State Variable equation rupture stress

1σ Maximum principal true stress

σ Cauchy stress tensor

θ1, θ2, θ3, θ4 Theta Projection Concept creep constants

ω Creep damage

ω& Creep damage rate

ρ Density at time t

1

Chapter 1 Introduction

1.1 Motivation

Pressure vessels have been experimentally shown to fail when exposed to accidental fire

loads. The failure can be highly destructive if the pressure vessel contains both a saturated liquid

and vapour at the time of failure as a boiling liquid expanding vapour explosion (BLEVE) could

occur (Birk et al., 2006). Failure occurs primarily due to high wall temperatures in the vapour

wetted region of the vessel. With a properly working and sized pressure relief valve (PRV), the

vessel can still fail due to wall strength degradation at high temperatures even though the vessel

pressure is within design limits (Birk et al., 2006). Experimental pressure vessel fire tests have

shown that thermally unprotected pressure vessels could fail in less than 30 minutes (Birk et al.,

2006). This is a topic of engineering interest since some pressure vessel design standards, such as

that for certain North American Rail Tank-Cars (CFR 49 Part 179, 2008; CGSB-43.147, 2005;

ASME,2001), require that pressure vessels resist failure for at least 100 minutes in defined fully

engulfing fires. Therefore, if a pressure vessel is exposed to an accidental fire, there exists the

possibility for a highly destructive explosion to occur in a relatively short amount of time. The

motivation for this work was to employ finite element analysis (FEA) in order to better

understand the structural failure of pressure vessels exposed to a variety of accidental fire loading

conditions.

1.2 Pressure Vessel Failure

This work deals with pressure vessels containing pressure liquefied gases such as

propane, and any pressure vessel discussed here should be assumed to be filled with propane.

Also, the work deals only with pressure vessels made of ASME SA 455 steel and all pressure

vessels discussed here should be assumed to be made of SA 455 steel. However, pressure vessels

2

made of other steels could also be analyzed using the same techniques presented in this work;

new material constants would need to be investigated.

When a pressure vessel is heated due to an accidental fire, its wall material weakens and

its internal pressure rises. These two effects push the tank towards failure. However, most

pressure vessels are equipped with a pressure relief valve (PRV) that is programmed to open in

order to release pressure from the vessel once the internal pressure has risen above a

predetermined value. Therefore, in the event of a vessel being exposed to an external heat source,

a race between the fire weakening the vessel material and the PRV emptying the vessel occurs.

Figure 1-1 shows a pressure vessel failure that occurred during an experimental fire test.

Figure 1-1: Experimental pressure vessel fire test (Birk private communication, 2008).

Depending on the fill level of a pressure vessel, a certain percentage of the vessel volume

will be occupied with saturated liquid and the remainder will be occupied with vapour. The

compressed liquid will occupy the bottom portion of the pressure vessel as it has a higher density

than the vapour. Data from fire tests (see for example Birk et al. (1994)) show that the liquid

wetted wall stays below about 130oC during a fully engulfing fire, while temperatures of the

3

vapour wetted wall can rise to anywhere between 550oC to 750oC depending on the intensity of

the fire as well as other external factors. The liquid wetted wall remains much cooler because the

liquid propane is much more efficient at removing heat from the vessel wall than the vapour is.

This is because liquid propane has a higher thermal conductivity, density and specific heat then

vapourous propane; i.e. liquid propane has much higher convective heat transfer coefficient than

vapourous propane. Phase change due to boiling of the liquid also enhances heat transfer from

the vessel wall to the liquid. When pressure vessels are exposed to engulfing fires, they

consistently fail at the location of peak wall temperature, somewhere in the vapour space wall

region.

Short-term, high-temperature failures are termed stress-rupture failures. Stress-rupture

includes two significant subcategories: short-term overheating and high-temperature creep

(Viswanathan, 1989).

Short-term overheating failures are driven by plastic deformation. In short-term

overheating, a component is exposed to excessively high temperatures to the point that material

yielding begins. Considerable deformation of the given component is observed during short-term

overheating in the form of reduced wall thickness and the failure surface resembles a knife edge

fracture. Due to the excessively high temperatures, material phase changes are common during

short-term overheating (Viswanathan, 1989).

High-temperature creep failures are driven by tertiary creep deformation. Temperatures

are high enough to result in relatively fast failure times but not high enough to cause material

yielding. Little or no reduction in wall thickness is observed during high-temperature creep;

however, significant and measurable creep deformation does occur. The fracture surface will

have thick edges since creep damage creates link ups of individual voids (Viswanathan, 1989).

4

Kraus (1980) experimentally observed that the quicker a tertiary creep dominated creep failure

occurred, the more ductile the rupture was. This indicates that there exists a cross over region

between high-temperature creep and short-term overheating in which both phenomena are acting

to deform the material.

It is important to note that a combination of these two failure modes can occur. For

example, temperature and stresses can be high enough to cause material yielding but not

structural failure by plastic deformation. In this case, after the material yields, high-temperature

creep would continue to occur until failure. In order to accurately predict pressure vessel failure

behaviour, both stress-rupture failure modes need to be incorporated into the numerical analysis

model. Therefore, the analysis model will compute high-temperature creep deformation while

checking for and if necessary, computing localized plastic deformation. This is an elastic-plastic-

creep deformation model.

From pressure vessel fire tests conducted by Birk et al. (2005) both stress rupture failure

modes were observed. Very rapid failures (<10 mins) in which peak wall temperatures were very

high (above 720oC) had knife edge fractures, indicating short-term overheating. Failures that

took longer and had lower peak wall temperatures (<680oC) were characterized by some necking

and rough failure edges, indicating a combination of short-term overheating and high-temperature

creep.

1.3 Objectives

In order to develop a pressure vessel finite element analysis failure model, three main

objectives were completed: SA 455 steel’s elevated temperature tensile and high-temperature

creep properties were measured, creep properties were extracted from experimental data and

5

validated for uniaxial finite element creep analysis, and a pressure vessel finite element analysis

model was developed.

In order to model the phenomena that govern the tank’s failure, SA 455 steel high

temperature material properties were needed. ASME Boiler and Pressure Vessel Code (ASME,

2004) provided some limited data on the temperature dependency of SA 455 steel’s mechanical

properties. However, it was impossible to get detailed true stress-true strain tensile data from this

ASME data. Moreover, there was no published creep rupture data for SA 455 steel. This lack of

high temperature data was due to the fact that SA 455 steel is not intended for high temperature

applications. However, for the accidental loading being investigated, these properties were

needed to conduct the analyses. Therefore, a series of mechanical property evaluation tests were

performed at Queen’s University.

Creep of steels is usually studied on the time scale of hundreds or thousands of hours.

Since stress rupture of pressure vessels occurring on the order of minutes was investigated, it was

not clear which creep damage model(s) would best suit this application. Thus, an extensive

literature review was conducted to choose creep damage models that showed potential to produce

accurate results for this work. To ensure that the chosen creep damage models were performing

adequately, finite element analyses were carried out using the creep constants extracted from the

experimental material data to predict the uniaxial creep behaviour observed in the lab. For these

analyses, the creep models were implemented into commercial FEA computer packages ANSYS

and ABAQUS via creep user subroutines. An in-depth comparison between experimental and

numerical analysis of the uniaxial creep behaviour of SA 455 steel was performed to validate all

derived creep constants.

6

The scope of this work was to model the behaviour of pressure vessels exposed to a range

of accidental loading conditions using concepts of solid mechanics. Therefore, the work was

conducted with constitutive equations that were macroscopic in nature (dealt with stresses and

strains averaged over representative volumes). Nevertheless, these macroscopic representative

equations were developed to reflect the microstructural mechanisms that govern the material

behaviour as best as possible.

Several finite element analysis models were developed using ANSYS and ABAQUS to

examine the stress rupture behaviour of various pressure vessels exposed to various loadings.

The effects that wall thickness, wall temperature, vessel internal pressure, local hot spots and

vessel dimensions have on pressure vessel failure were investigated.

7

Chapter 2 Literature Review

The literature used to gain background information on high temperature structural failure

of pressure vessel steel is split into the following three areas: mechanical testing, numerical creep

failure prediction and pressure vessel failure analysis.

2.1 Mechanical Testing

2.1.1 Tension Tests of Metallic Materials

The American Standard of Testing Materials (ASTM) standard E 8M-98 (ASTM, 1998)

presents the needed information to conduct a uniaxial tensile test on a metallic material. The

following is a brief summary of key points presented in the standard that apply to the mechanical

testing that was performed. For a plate-type specimen with pin ends, the standard specifies that

the gage length must be at least 4 times the thickness of the sample. During the tensile tests, a

strain rate between 0.05min-1 and 0.5min-1 must be used. An extensometer should be used to get

accurate strain measurements during the room temperature tensile test. The yield strength is

determined by the offset method; finding the intersection of the tensile test curve with the 0.2%

strain offset of the elastic portion of the tensile test curve. The ultimate tensile strength is found

by dividing the maximum load achieved in the tensile test by the original sample cross sectional

area. Elongation is the percentage change in gage length achieved during the testing. The gage

length is to be measured to the nearest 0.05mm before and after the tensile test in order to

calculate the elongation.

8

2.1.2 Elevated Temperature Tension Tests of Metallic Materials

The American Standard of Testing Materials (ASTM) standard E 21-92 (ASTM, 1998)

presents the needed information to conduct an elevated temperature uniaxial tensile test on a

metallic material. The following is a brief summary of key points presented in the standard that

apply to the mechanical testing that was performed. All of the points mentioned above in the

summary of standard E 8M-98 apply to elevated temperature tests as well. In addition, the

following criterion must also be met. For a gage length of 2” or less, at least two thermocouples

must be attached to the specimen, one near each end of the gage length. For a test temperature up

to 1000oC a difference of ± 3oC is allowed between the indicated test temperature and the

nominal test temperature, above 1000oC, ± 6oC is allowed. Temperature overshoots during initial

sample heating should not exceed these limits. Unless otherwise stated, the time of holding at test

temperature prior to the start of a test should not be less than 20 minutes. This time is given to

ensure that the specimen has reached equilibrium and that the temperature can be held between

the limits mentioned above.

2.1.3 Creep Rupture Tests of Metallic Materials

The American Standard of Testing Materials (ASTM) standard E 139 – 96 (ASTM,

1998) presents the needed information to conduct creep rupture tests on a metallic material. The

following is a brief summary of key points presented in the standard that apply to the mechanical

testing that was performed. A creep rupture test is defined as “a test in which progressive

specimen deformation and the time for rupture are measured”. These measured values depend on

test load and temperature. The specimen specifications mentioned in standard E 8M-98 apply to

creep rupture testing as well. For a gage length of 2” or less, at least two thermocouples must be

9

attached to the specimen, one near each end of the gage length. For a test temperature up to

1000oC a difference of ± 2oC is allowed between the indicated test temperature and the nominal

test temperature, above 1000oC, ± 5oC is allowed. Temperature overshoots during initial sample

heating should not exceed these limits. Unless otherwise stated, the time of holding at the test

temperature prior to the start of the test should not be less than 1 hour. This time is given to

ensure that the specimen has reached equilibrium and that temperature can be held between the

limits mentioned above. During testing, temperature overshoots are much more of a concern than

a temperature drop and any test that has a temperature rise above the limits specified above

should be rejected. Over temperature may considerably accelerate creep and alter test results

whereas, low temperatures usually will not damage the material in this fashion and if exposure

time at low temperatures is not significant the test results may be kept. Strain readings must be

made at least every 24 hours or 1% of the estimated duration of the test. The load should be

applied such that shock loads or overloading due to inertia is avoided. The following data should

be included in the report from a creep rupture test: time to rupture, test temperature, nominal test

stress, elongation, minimum creep rate and time to various (0.1%, 0.2%, 0.5%, 1.0%, 2.0%,

5.0%) total strains; for a full list of data to be reported from a creep rupture test, see ASTM

standard E 139 – 96 (ASTM, 1998).

2.2 Numerical Creep Failure Prediction

Creep failure can be numerically predicted using a creep damage model. The idea of

quantifying a creep damage parameter was originally introduced by Kachanov (1960). The main

purpose of a creep damage model is to predict failure in a component that is exposed to creep

loading conditions. Most of the models work similarly when coupled with finite element

analysis. A damage parameter is integrated over time at each node or integration point of the

10

finite element analysis model. That node or integration point is considered to have failed when

the damage parameter reaches a certain critical value (usually 1). Creep deformation and creep

damage models commonly used in research and industry are reviewed in the following

subsections.

2.2.1 Creep

The important dates in the development of the mathematical modeling of creep were

(Lemaitre and Chaboche, 1985):

• 1910, Andrade’s law of primary creep.

• 1929, Norton’s law which linked the rate of secondary creep to stress.

• 1934, Odqvist’s generalization of Norton’s law to the multiaxial case.

Collins (1993), among others, provides comprehensive information on creep phenomena,

creep behaviour and design for creep.

Creep deformations are permanent, time-dependent and are considered to develop and

accumulate in accordance with the concepts of incompressible material deformation. Time

dependent implies that the rate of deformation is significant enough that it must be considered in

the engineering assessments pertaining to the viability of a structure. Continued creep

deformation occurs under constant load and generally, as a rule of thumb for metals and metal

alloys, it must be accounted for if temperatures are above 40%-50% of the homologous material

melting temperature; that is temperature in degrees Kelvin. Extended time at elevated

temperatures can act as a tempering process, reducing material strength.

The melting point of a generic medium carbon steel is about 1450oC (Reed-Hill and

Abbaschian, 1992). Therefore, creep of medium carbon steels should be taken into account at

temperatures above about 420oC – 590oC. Note, that there exists no published data on the

11

melting temperature of SA 455 steel but that its carbon content classifies it as a medium carbon

steel (0.33%). However, SA 455 steel also has a significant amount of Manganese (1.14%)

which might affect its melting temperature. From this information and from the known

temperature distributions of pressure vessels exposed to accidental fires, the creep rupture

properties of SA 455 steel were measured in the range of 550oC – 720oC.

Although creep is a plastic flow phenomenon, creep rupture typically occurs without

necking and can be catastrophic. However, for the case of relatively fast creep ruptures, Kraus

(1980) has experimentally observed ductile failure surface; indicating a combination of creep and

plastic deformation.

Good creep resistance features for materials include:

• Metallurgical stability under long-time exposure to elevated temperatures;

• Resistance to oxidation and corrosive media; and

• Larger grain size since this reduces the area of grain boundaries, where much of the creep

damage is perceived to take place.

The rate at which a material creeps depends on numerous aspects, which include:

• Material microstructure

• Temperature

• Stress

• Mode of loading (uniaxial, multiaxial, etc.)

• Neutron flux (nuclear applications)

The relationship between temperature, stress and failure time is plotted in Figure 2-1,

where it is seen that higher temperatures will cause reduced failure time and increased stress will

also cause reduced failure time. Kraus observed that failures on the left portion of the graph are

12

ductile, whereas failures on the right portion are brittle, using reduction in area as a measure of

ductility.

Figure 2-1: Creep failure time dependence on temperature and stress (Kraus, 1980).

Note that this work is interested in failures that occur in less then 100 minutes, i.e. on the

extreme left of Figure 2-1.

The most common way to obtain creep data is from uniaxial creep rupture tests in which

a tensile specimen is loaded under constant load to failure. Due to the nature of this test, the time

required can be long. A less common creep test is the stress relaxation test under which the

specimen is loaded to a constant strain and the decrease of the load is monitored. This test is

somewhat faster. Displacement (converted to strain), load (converted to stress) and time to failure

are all recorded from a creep rupture test. Figure 2-2 shows the typical creep curve of a metallic

material. The affect that temperature and stress have on a typical creep curve was plotted in

Figure 2-3 and 2-4.

13

Figure 2-2: Creep strain vs. time; typical plot generated from a creep rupture test (Kraus,

1980).

14

Figure 2-3: Temperature dependence of a typical creep curve (Kraus, 1980).

Figure 2-4: Stress dependence of a typical creep curve (Kraus, 1980).

15

Examination of Figure 2-2 indicates that there are 3 distinct phases of creep: primary,

secondary (or steady state) and tertiary. It is important to note that the shape of this graph is

strongly affected by test temperature and stress, see Figure 2-3 and 2-4. In the primary phase of

creep, creep deformation decelerates until the minimum creep rate is reached. The amount of

creep deformation and thus creep damage accumulated during this phase is often neglected as it is

considered small compared to the total creep strain at failure. The secondary phase of creep is

often called “steady state creep” since during this phase the creep rate is constant. During the

tertiary phase of creep, deformation accelerates. Depending on the exposure temperature, applied

stress and the material microstructure, this phase will vary significantly.

Stress relaxation is another interesting creep deformation phenomena. Stress relaxation

occurs when strain is held fixed during creep loading conditions and the typical behaviour is

shown in Figure 2-5.

Figure 2-5: Effect of stress level on relaxation at constant temperature (Kraus, 1980).

16

2.2.2 Kachanov/Rabotnov (Empirical based Continuum Damage Mechanics)

Kachanov was the first to develop the idea of continuum damage mechanics. His work

was revised by Rabotnov so that, among other things, the damage parameter varied from 0 in an

undamaged state to 1 in a failed state. The equations developed by Kachanov and revised by

Rabotnov have been widely used. Some modifications to the original equations have been

proposed by a number of authors who were attempting to increase analysis accuracy for their

specific applications. Hyde, Becker and Sun (1993-2006) (among others) have had success

predicting creep failure in a variety of applications using the Kachanov type equations in forms

they refer to as the One-State Variable equations and Three-State Variable equations.

The One-State Variable equations were derived empirically to match experimental

tertiary creep data results and are presented in Section 3.0.

Hyde (1998) presents in detail the steps required to extract the needed One-State Variable

creep constants from experimental creep rupture data. Becker et al. (2002) have shown the One-

State Variable creep equations to be accurate using numerous benchmarks. Hyde et al. (2006),

Sun et. al (2002) and Hyde et. al (1999) have shown that the One-State Variable equations can

accurately predict creep behaviour of various steel pipes exposed to creep loading conditions.

Figure 2-6 shows the uniaxial validation that Hyde et al. (2006) performed to ensure that the One-

State Variable equations and Three-State Variable equations could predict creep behaviour of Bar

257 steel.

17

Figure 2-6: (a) Tested and fitted uniaxial creep strain curves of the Bar 257 steel at 650 oC for the one-state variable equation. (b) Tested and fitted uniaxial creep strain curves of the Bar 257 steel at 650 oC for the three-state variable equation (Hyde et al., 2006).

Due to its simplicity and reliability, the One-State Variable form of empirical based

Continuum Damage Mechanics Technique was one of the creep damage models chosen to

numerically model the creep behaviour of SA 455 steel in this work.

2.2.3 MPC Omega Method

The MPC Omega method was developed by the Materials Properties Council and was

presented by Prager (1995). The MPC Omega method allows one to predict remaining life and

creep strain accumulation of the component being assessed.

The method is based on the idea that the current creep strain rate, along with a brief

history of creep strain rates, provides enough information to be able to predict creep behaviour of

a component both for past creep exposure and future creep exposure. The sole parameter of the

method, which is derived experimentally, is Ωp and it can be found by plotting the natural

18

logarithm of true creep strain rate vs. true creep strain from creep rupture test data (Figure 2-7);

the slope of this straight line gives the parameter Ωp. If a straight line for this plot is not obtained

from the data, then another creep damage model should be sought. It is important to note that Ωp

is dependent upon temperature, stress and mode of loading. Generally, Ωp (Prager, 2000):

• Increases with decreasing stress

• Increases with decreasing temperature

• Increases with increasing multiaxiality

• Is far less sensitive to stress and temperature than strain rate

A large Ωp, 30, 50 or even 200 or more, indicates that most of the service life was spent at

very low strains. In the final stages of life, strain rate accelerats rapidly to failure. High Omega

behaviour may be due to creep softening or brittleness (Prager, 2000).

A low Ωp, would indicate that the service life was spent mostly in a tertiary state of creep

strain. However the acceleration of the creep strain is not as prominent as that in the final stages

of life for a material with high Omega values (Prager, 2000).

19

Figure 2-7: Deriving Ωp from experimental data; strain rate has units hr-1 (Prager, 1995).

From creep rupture data one can create a table of the stress and temperature dependent Ωp

values as shown in Table 2-1.

Table 2-1: Experimentally derived Omega values for carbon steel (Prager, 1995).

20

The MPC Omega method is included in the American Petroleum Institute Recommended

Practice 579 on Fitness-for-Service as a reliable method of predicting creep failure in high

temperature applications with commonly used metals (API, 2000). One advantage of this method

is that the Fitness-for-Service report includes all the needed material constants for several

common high temperature application metal alloys. The inclusion of the MPC Omega method

into the API Recommended Practice shows that it is a creep failure prediction method that is

trusted by many practicing engineers.

Prager (1996) has shown that the MPC Omega method can accurately predict high

temperature tube life and creep crack growth curves. Kwon et. al (2007) have shown that the

MPC Omega method can accurately predict creep life of heater tubes. Evans (2004) has

compared MPC Omega method’s creep property prediction accuracy against other creep damage

techniques. Evans concluded that the MPC Omega method produced relatively accurate results

compared with other creep life prediction techniques (Theta Projection Concept, CRISPEN and

Kachanov) and that the MPC Omega method excelled at predicting minimum creep rates.

The MPC Omega method has been proven to be a reliable technique for predicting creep

rupture by several researchers and by the American Petroleum Institute. Therefore, the MPC

Omega Method was one of the creep damage models chosen to numerically model the creep

behaviour of SA 455 steel in this work.

The empirically derived equations of the MPC Omega Method are presented in Section

3.0.

2.2.4 Theta Projection Concept

The Theta Projection Concept was developed by Evans and Wilshire (1984) to accurately

extrapolate creep rupture data. In order to accomplish this, a large number of constants

21

(minimum 17) need to be derived from experimental creep rupture data. It is important to note

that the Theta Projection Concept does not model secondary state creep, it only models primary

and tertiary creep and views secondary creep as simply a transition between the other two.

The Theta Projection Concept attempts to predict total true creep strain using the

following empirically derived equation:

( ) ( )11 4231 −+−= − ttc ee θθ θθε (Equation 2-1)

where, cε is creep strain, t is time, θ1 and θ3 define the extent of creep strain and θ2 and θ4

quantify the curvatures of the primary and tertiary stages of creep respectively. An advantage of

this method is that once the stress and temperature dependencies of the four θ functions are

determined, any creep rupture parameter can be easily computed (Brown et al. 1986).

Empirically, the following function was developed to quantify the stress and temperature

dependencies of θi:

TT iiiii σδγσβαθ +++=log (Equation 2-2)

where, iiii and δγβα ,, are material constants used to define the stress and temperature

dependency of iθ .

Failure is determined by exhaustion of ductility, a failure strain, which is dependent upon

stress and temperature and extrapolated using the Theta Projection Method. The temperature and

stress dependency of the failure strains is given by the following equation:

TdcbTaf σσε +++= (Equation 2-3)

where a new set of a, b, c and d material constants need to be determined, bringing the

total number of constants required to 20.

A drawback of the Theta Projection Method is that failure is based on a creep failure

strain. Hyde (1998) suggests that in most cases, experimental rupture times are much more

22

reliable and less prone to variability than rupture strains. Depending on the material and testing

conditions, the creep failure strain can show significant variance. However, for some materials

under certain testing conditions the creep failure strain can be almost constant (Brown et al.

1986). Therefore, when using the Theta Projection Method an investigation of failure strain

variability is important.

Researchers have applied the Theta Projection Concept to successfully predict creep

behaviour of various components. Evans and Whilshire (1986) and Brown et al. (1986)

successfully modeled creep of turbine blades using the Theta Projection Concept. Loghman and

Wahab (1996) modeled creep of thick tubes using the Theta Projection Concept. Law et al.

(2002) modeled creep of thick-walled pressure vessels with a through-thickness thermal gradient

using the Theta Projection Concept. Figure 2-8 shows the stress redistribution that occurred in

the pressure vessel due to creep deformation. Law et al. were not concerned with analyzing

pressure vessel failure. Rather, their main objective was to investigate the creep deformation

behaviour of a pressure vessel with a through-thickness temperature gradient. Figure 2-9 shows

that the surface of the pressure vessel with the higher temperature will, as expected, creep faster.

Figure 2-8: Theta projection modelling of +5oC thermal case, von-Mises stresses (Law et al., 2002).

23

Figure 2-9: Variation of creep rate at inner and outer walls for +5oC case (Law et al., 2002).

When modeling creep deformation, i.e. when failure is not being predicted, the Theta

Projection method has been shown by many researchers to perform reliably. However, when

attempting to numerically predict creep failure, the failure time is much more important than

failure strain. Therefore, it was decided that a creep damage model based on failure times or

creep strain rates was more appropriate for this work, so the Theta Projection Concept was not

used.

2.2.5 Physically Based Continuum Damage Mechanics

There are several mechanisms by which a microstructure can degrade in service.

Therefore, it is unlikely that a single equation, like those used in empirically based continuum

damage mechanics, would be able to accurately predict creep behaviour. Thus, physically based

continuum damage mechanics was introduced by Ashby and Dyson (1984). Physically Based

CDM is much more complex than empirical CDM and has many more parameters that need to be

quantified. It is beyond the scope of this literature review to introduce all of the physically based

24

continuum damage mechanics equations. Due to its complexity this creep damage model was not

used in this work. However, this is a powerful creep failure prediction tool utilized by many

researchers and thus a brief overview is provided.

According to Dyson (2000) the following four steps need to be completed to implement a

physically based CDM model:

• identification of each applicable damage mechanism;

• definition of a dimensionless (damage) variable for each mechanism;

• incorporation of each variable within a kinetic equation for creep;

• development of an evolution equation for each variable.

Several microstructural parameters are incorporated into the physically based continuum

damage mechanics equations. For example, the following damage variables could be

incorporated into the model: (i) stress redistribution around hard regions; (ii) a progressive

multiplication of the mobile dislocation density; (iii) coarsening of the strengthening particles;

and (iv) nucleation-controlled, constrained grain boundary cavitation.

Dyson (2000) obtains the values of these parameters by initially choosing a set of values

based on theoretical expectations. Creep curves are then computed by integrating the creep strain

rate equations at different initial stresses and temperatures using a fifth-order Runge-Kutta

numerical method with adaptive step-size control. Using the software CREEPSIM-2,

convergence towards the “best” parameter set was observed by repeated systematic manual

changes of parameters within the imposed physical bounds.

Physically based CDM has been successfully applied to predict creep behaviour in a

variety of components made of various engineering alloys by Ashby et al. (1984), Dyson et al.

(1989) and Hayhurst et al. (1994) among others. Yang et al. (2004) performed creep failure

25

analyses on variously notched tubes using Physically Based Continuum Damage Mechanics;

Table 2-2 summarizes their findings.

Table 2-2: Predicted failure times for plain and notched tubes with experimental data for comparison. Test conditions: 565°C, 36.6 MPa. (Yang et al., 2004)

Yang et al.’s results illustrate the superior predictive capability of Physically Based CDM

methodology compared with more conventional methods. It should be noted that for creep failure

analyses, predictions within about 20% of measured failure are considered to be excellent. They

attribute this significant difference in accuracy to the fact that:

• Conventional methods do not allow for stress redistribution as a result of creep

deformation.

• Physically based CDM defines failure as occurring when the component can no

longer carry equilibrium loads as a result of damaged elements.

• The ASME approach defines failure based on surface flaws.

For the numerical creep failure analysis of pressure vessels, the concern is not with

modeling the individual mechanisms of creep deformation, but rather with developing a model

that would reliably predict the global material behaviour associated with high-temperature failure

of steel. Also, Physically Based Continuum Damage Mechanics would make computation and

material property testing times much higher then needed for this application. Thus, it was

decided that the Physically Based Continuum Damage Mechanics would over complicate the

analysis and was therefore not practical for use in this work.

26

2.2.6 Robinson’s Life Fraction Rule

Robinson’s Life Fraction Rule was originally proposed to estimate creep lifetimes of

components operating under variable temperatures during service (Robinson, 1938). The key

advantage of this method is that, due to its simplicity, it can be applied to pretty much any creep

model analysis to predict component lifetime. However, according to Dyson (2000), a major

drawback of the method is that under conditions of varying stresses it will generally not produce

accurate results. This can be explained with the same argument that is used to show why creep

strain equations based on time hardening are not as accurate as those based on strain hardening

when dealing with varying stresses (Kraus, 1980); see Section 3.1.2 for more details.

The creep damage of the component is given by Equation 2-4, and is integrated over its

lifetime with respect to stress and temperature.

),( σω

Ttt

r

s= (Equation 2-4)

where, ω is creep damage, ts is time in service, tr is rupture time, T is temperature and

σ is stress.

Since the damage values over each time period are assumed to be independent of each

other, failure will occur when ∑tω reaches a value of 1.

Due to the fact that Robinson’s Life Fraction Rule has been shown to produce unreliable

results under varying stress conditions, it has not been used much recently by researchers. For

this same reason it was not used for this work.

27

2.2.7 Larson-Miller

The Larson-Miller parameter presented by Larson and Miller (1952) is a quick way to

organize and extrapolate experimental creep rupture data so that it can be plotted as a single

function on a single graph. The Larson-Miller parameter combines test temperature and rupture

time in a single parameter, the Larson-Miller parameter, so that it can be plotted against stress. It

should be mentioned that this method, on its own, cannot be coupled with finite element analysis

to evaluate creep rupture in components.

The function used to combine test temperature and rupture time is:

)log( rMLML tCTP += −− (Equation 2-5)

where, PL-M is the Larson-Miller parameter, T is absolute temperature (oR), CL-M is the

Larson-Miller constant and tr is the rupture time (hrs).

For example, stress is plotted versus the Larson-Miller Parameter in Figure 2-10 below.

A polynomial of first or second order is generally found to best describe the PL-M

function. This function can be extrapolated past the experimental data in order to obtain creep

information outside of test condition boundaries. Because the Larson-Miller technique depends

on a “common” constant (CL-M ≅ 20), it can easily be used for relative assessments of creep

behaviour of various materials. Note that choosing a value of CL-M=20 is standard practice, when

no other information indicating otherwise is present.

28

050

100150200250300350400

24000 29000 34000 39000

Larson-Miller Parameter PL-MS

tress

[MP

a]

Figure 2-10: Larson-Miller plot for SA 455 steel.

It is possible to calculate the optimal value of the Larson-Miller constant for a specific set

of creep rupture data. This process was outlined by Kraus (1980). It is important to note that in

order to calculate the Larson-Miller constant, the following units must be used: degrees Rankine

for temperature and hours for time. Also, at least two sets of data are needed that each have at

least two creep rupture tests that were conducted at different temperatures but with the same

stress.

The Larson-Miller technique on its own cannot be integrated over time to account for

varying operating temperatures and stress conditions. Since most applications will see varying

temperature and stress histories, the Larson-Miller technique cannot be coupled with a finite

element analysis on its own in order to predict creep rupture. However, one could combine

Robinson’s life fraction rule with the Larson-Miller technique and implement this combined

model into a finite element analysis to predict creep failure. The Larson-Miller technique would

provide Robinson’s rule with the stress and temperature dependent life times needed to calculate

life fractions consumed at each node, element or integration point of a finite element model.

29

These fractions would then be summed at each node, element or integration point over the life of

the component to determine both failure times and failure locations.

The Larson-Miller technique was used to organize SA 455 steel’s stress rupture data.

However, as discussed, it was not used to numerically predict creep behaviour.

2.3 Pressure Vessel Failure Analysis

Fire tests of pressure vessels have been conducted by many researchers (Birk,

Cunningham, Ostic & Hiscoke 1997, 2004; Droste & Schoen 1988; Moodie, Cowley, Denny,

Small & Williams 1988, among others) in order to achieve a more detailed understanding of high

temperature pressure vessel failure mechanics. These tests collected vast amounts of data

including, vessel wall temperatures, internal pressure and liquid and vapour temperatures in the

vessel. The data was analyzed to make conclusions on how various test parameters affected

failure times of the pressure vessels as well as whether or not a boiling liquid expanding vapour

explosion (BLEVE) occurred. Results from experimental pressure vessel fire tests provide

valuable information regarding vessel temperature distribution, internal pressure history and

failure time.

All the reviewed work was similar in that great detail was given to modeling the thermal

response of the vessels but the stress analysis conducted was quite simplistic. Thus, the current

work built upon past work conducted by others by utilizing the thermal response data collected

and analyzed to input thermal loads as temperatures at nodes into finite element analyses that

conducted detailed stress analyses of the vessels.

30

2.3.1 Thermal Response of a Pressure Vessel in an Accidental Fire

Droste and Schoen (1988) fire tested unprotected and thermally protected pressure

vessels containing liquid propane gas in fully engulfing fire conditions. The goal was to examine

the safety margins of unprotected pressure vessels. The pressure vessels were equipped with

several NiCr/Ni thermocouples and pressure measurement devices. They observed that the

temperature values of the tank wall in the liquid space of the vessel were similar to those of the

compressed liquid propane’s temperature. Also, they observed that the vessel temperatures in the

vapour space were much higher and that the maximum tank wall temperature was always

measured at the top of the horizontal cylindrical vessel. A typical temperature distribution history

of a pressure vessel thermally protected with an insulation jacket is shown by Droste and Schoen

(1988) in Figure 2-11.

Figure 2-11: Thermally protected pressure vessel shell temperatures (Droste and Schoen,

1988).

31

Moodie et al. (1988) carried out five fire tests on commercial propane pressure vessels

with initial fill levels of 22% to 72%. The main objective of this work was to gain a better

understanding of how the pressure vessels heated up in a fully engulfing fire. To avoid failure,

the fires were allowed to burn until peak vessel wall temperature reached about 600oC. They

observed that the outer liquid-wetted wall temperatures ranged from 50oC to 140oC and that a

through thickness temperature gradient of about 20oC was present in the liquid-wetted wall. They

also observed that all the vapour-wetted wall temperatures behaved similarly, increasing rapidly

once the fire had become established, but they leveled off once the pressure relief valve started

venting the vessel; see Figure 2-12. Moodie et al. also measured a “frothing” region, a region in

which wall temperatures sharply changed from hot vapor-wetted wall temperatures to much

cooler liquid-wetted wall temperatures. They reported that temperature measurements just above

the initial liquid level indicated that some upwelling of liquid and frothing had occured once the

fire was established; this was expected since the liquid will expand when heated and the liquid

surface will contain vapour bubbles as it boils.

Figure 2-12: Thermally unprotected peak vessel wall temperatures; various initial fill levels

(Moodie et al., 1988).

32

The thermal response behaviour of pressure vessels in accidental fires as observed by

Droste et al. and Moodie et al. was also observed by Birk et al. (1994, 2006) during many of their

experimental vessel fire tests, see for example Figures 2-13 and 2-14. The “04-3” in figure

captions refers to testing conducted in 2004 and the test number.

Figure 2-13: Wall thermocouple layout for test 04-3 (Birk et al., 2006).

33

Figure 2-14: Thermally protected pressure vessel wall temperature, test 04-3 (15% area insulation defect); wall 12, 13 and 14 are on outer surface on thermal protection (Birk et al.,

2006).

2.3.2 Pressure Vessel Failure due to Accidental Fire Loading

The pressure vessels tested by Droste and Schoen had a fill level of 50%. They

postulated that there were two reasons for thermally unprotected pressure vessels to rupture in a

fire accident:

• Increase of internal pressure caused by the temperature rise of the compressed

propane and insufficient pressure relief of the safety valve.

• Decrease of the vessel’s structural strength caused by an increasing vessel wall

temperature and a corresponding drop of the yield and tensile strength.

34

Droste and Schoen calculated the bursting pressure of the vessel under the assumption

that rupture will occur when the material’s temperature dependent yield strength is exceeded.

The comparison of calculated values to test results in Figure 2-15 shows that the vessels have a

reasonable margin of safety and that the assumption that rupture occurs when the material’s yield

strength is exceeded is conservative.

Figure 2-15: Comparison of calculated tank bursting pressures and test results

(Droste and Schoen, 1988).

35

Droste and Schoen observed that internal pressure rise was not stopped or significantly

delayed by the start of discharge of the pressure relief valve. From this work it was concluded

that the thermal insulation design tested was able to prevent vessel failure even in a 90 minute full

fire engulfment.

Moodie et al. (1988) concluded from their work, among other things, that an increase in

wall temperatures in the vapour space and the long term behaviour of the vessel internal pressure

during PRV operation would ultimately determine the integrity of the vessel structure.

Birk (2006) also found that pressure vessel rupture by fire impingement is due primarily

to pressure induced stress and high wall temperatures in the vapour space wall region.

2.3.3 Computer Modeling of Pressure Vessel Failure in Accidental Fire Loading

In order to better understand the mechanisms involved in high temperature pressure

vessel failure there have been numerous computer codes developed to model the response of

pressure vessels exposed to accidental fires. Birk (1988) presented a computer model, the Tank-

Car Thermal Computer Model (TCTCM), which could simulate the thermal response of a

pressure vessel and its lading to external fire impingement and also predict vessel failure. The

computer model was developed as a tool to study the effectiveness of various thermal protection

systems of pressure vessels. The computer code could model a fully engulfing fire or a local hot

spot. It could account for various protection systems such as pressure relief valves, thermal

insulation, radiation shielding, temperature sensing relief valves and internal dissipating matrices.

TCTCM models the pressure vessel as a two-dimensional representation of a circular cylindrical

pressure vessel (axial gradients and end effects are not accounted for). The model was made up

of a series of submodels that simulated the following processes:

• Flame to vessel heat transfer,

36

• Heat transfer through the vessel wall and associated coverings,

• Interior-surface to lading heat transfer,

• Thermodynamic process within the vessel,

• Pressure relief device operating characteristics,

• Wall stresses and material property degradation, and

• Tank failure.

Rupture of the pressure vessels was predicted by TCTCM as a function of vessel

geometry, internal pressure, vessel wall temperature and temperature dependent material strength.

It was observed that if very high wall temperatures were achieved, then the vessel ruptured even

if the relief valve maintained the vessel internal pressure well below the normal burst pressure.

TCTCM is capable of modeling the physical phenomena which govern heat transfer

between the fire and the pressure vessel and its contents, see Figure 2-16. Despite a highly

simplified calculation of vessel stress, the computer model was capable of accurately predicting

failure times of various experimental pressure vessel fire tests, see Figures 2-17 and 2-18.

37

Figure 2-16: Predicted and measured tank wall temperatures vs time from fire ignition for

upright uninsulated fifth-scale tank-car exposed to an engulfing fire (Birk, 1988).

Figure 2-17: Predicted tank wall stresses and material strength vs time from fire ignition for

upright uninsulated fifth-scale tank-car exposed to an engulfing fire (Birk, 1988).

38

Figure 2-17 shows that for some analyses the Tank-Car Thermal Computer Model was

not always in perfect agreement with experimental observations. Birk (1988) indicated that the

disagreement between predictions and the test could be caused by a number of factors such as

flaws in the tank wall, poor welds, or unmeasured hot spots in the tank wall (due to limited

thermocouple locations). Figure 2-18 shows excellent agreement between predicted failure time

and experimentally measured pressure vessel failure time.

Figure 2-18: Predicted tank wall stresses and material strength vs time from fire ignition for upright uninsulated full-scale tank-car exposed to an engulfing fire (Birk, 1988).

The computer model, TCTCM, developed by Birk (1983) was used to assess the fire

survivability of aluminum 33.5 lb propane cylinders vs steel propane cylinders. The study

showed that the aluminum cylinders would fail in about eight minutes in an engulfing fire and

39

that the steel cylinders would empty before failure occurred. Although no direct comparisons

were able to be made to experimental findings, accident data supports this finding. This work by

Birk (2003) adds to the reliability of the Tank-Car Thermal Computer Model.

Birk (2005) updated the Tank-Car Thermal Computer Model to accomplish the

following:

• Make the code quasi 3D so that it can account for a wide range of possible

thermal protection defect locations and geometries.

• Include graphical output features to confirm defect geometries.

• Refine lading thermal model to account for liquid temperature stratification and

its effect on tank pressure.

• Refine lading thermal model to include effects of defect position on tank

pressurization.

• Refine PRV model to include real-world PRV operating characteristics (such as

spring softening).

• Refine vapour space and PRV model to account for liquid entrainment into PRV

at very high fill levels.

• Refine tank failure model to include high-temperature stress-rupture.

• Ensure that the code runs in a reasonable length of time (<2 hours for the

simulation of a 100 minute fire scenario).

This updated computer model, Tank2004, was validated by Birk et al. (2006) against

experimental fully engulfing fire tests of pressure vessels with thermal protection defects, see

Table 2-3. The structural failure analysis was conducted using Robinson’s Life Fraction Rule

with the assumption that the effective failure stress is dominated by von Mises stress. The von

Mises stress used for these analyses is simplified as being a function of vessel dimensions and

40

internal pressure. Therefore, creep stress redistribution is ignored by the Tank2004 computer

model when calculating stress. The updated computer model produced good agreement with

experimental results and it was concluded that the assumption that von Mises stress drives the

tank rupture is reasonable.

However, as the size of the hot spot decreased, the Tank2004 computer model was found

to be less and less accurate. This was due to the fact that, as the hot spot size decreases creep

stress redistribution becomes increasingly prominent. Therefore, in order to accurately predict

high temperature failure of pressure vessels with hot spots, the stress state of the pressure vessel

needs to be completely modeled. The finite element analysis method is commonly used to

accurately compute the stress state of components numerically (among many other things).

Therefore, to improve upon Tank2004’s calculation of structural failure, the use of FEA is the

obvious choice to completely model the stress state and high temperature structural behaviour of

pressure vessels exposed to hot spots and fully engulfing fires.

Table 2-3: Summary of Tank2004 stress rupture predictions compared to actual failure times; 500 gallon SA 455 steel pressure vessels (Birk et al., 2006).

41

Chapter 3 Theory

3.1 High Temperature Steel Failure

Short-term, high-temperature failures are termed stress-rupture failures. Stress-rupture

includes two significant subcategories: short-term overheating and high-temperature creep

(Viswanathan, 1989).

During short-term overheating, temperatures and stresses are high enough to cause plastic

deformation. Since short-term overheating failures are driven by plastic deformation they are

considered to be time-independent. Section 3.1.1 provides a review of the basic theoretical

concepts of plasticity.

During high-temperature creep, temperatures and stresses are high enough to cause

tertiary creep deformation that will lead to relatively fast failure times. Note that the temperatures

and stresses are not high enough to cause material yielding. Since high-temperature creep

failures are driven by tertiary creep deformation they are considered to occur as time-dependent

phenomena. Section 3.1.2 provides a review of the basic theoretical concepts of creep.

It is important to note that short-term overheating and high-temperature creep can occur

simultaneously to cause failure. For example, stresses and temperatures can be high enough to

cause material yielding, but not failure by plastic deformation. In this case high-temperature creep

would continue to occur until failure. Another example is that if a material was deforming due to

high-temperature creep, it may become unstable as it nears failure. This instability can lead to

enough wall reduction to cause local plastic deformations as failure occurs. This type of

“combined” failure is often observed in experimentation (Kraus, 1980). Short-term overheating

produces a sharp knifed edge fracture face, while high-temperature creep produces a rough, thick

42

fracture face. However, in experimental fire tests of pressure vessels, fracture faces with wall

reductions of about 50% are common (Birk et al., 2006). This fracture face cannot be categorized

as either short-term overheating or high-temperature creep and thus it is concluded that it reflects

a combination of the two failure modes.

3.1.1 Plasticity

Plasticity can be defined as “that property that enables a material to be deformed

continuously and permanently without rupture during the application of stresses exceeding those

necessary to cause yielding of the material” (Mielnik, 1991).

The engineering stress-strain curve diverges from the true stress-strain curve as plastic

deformation commences. Therefore, when analyzing plastic deformation, we must use true

stress-stain data. The relationships between engineering and true stress and strain are shown in

Equations 3-1 and 3-2. Engineering stress and strain are calculated using the assumption that

original cross sectional area and original length are almost identical to current cross sectional area

and length. For metals, this is found to be a valid assumption while strains are less than about

0.2%. Therefore, for large strain problems, true stress and true strain must be used to accurately

model material behaviour.

)1( nn εσσ += (Equation 3-1)

)1ln( nεε += (Equation 3-2)

where, σ and ε are true stress and strain respectively and σn and εn are engineering stress

and strain.

43

The theory of plasticity is used to relate stresses and strains once the material has been

stressed past its yielding point. The yield point of a material is often defined by the 0.2% offset

stress, see Figure 3-1.

Figure 3-1: 0.2% offset used to define tensile yield strength (Hibbeler, 2003).

It is evident that a major difficulty in relating stress and strain during plastic deformation

is that the relationship between the two is no longer linear. Therefore, some assumptions are

made to model plastic behaviour. The von Mises theory of plasticity is most often used to model

plasticity in ductile metals. The key assumptions made for von Mises plasticity are (Mielnik,

1991):

• The material is homogeneous

• The volume remains constant, i.e, ΔV/ V and the sum of the plastic strain increments is

zero, or

0321 =++ ppp ddd εεε (Equation 3-3)

for plastic deformations where pε represents true plastic strain.

• A hydrostatic state of stress does not influence yielding.

44

Plasticity theory is composed of three parts that are needed to relate stress and strain during

plastic deformation: a yield criterion, a flow rule, and a hardening rule. The general theory has

many unique forms that were created to fit various sets of experimental data for various materials

and loading conditions.

The Yield Criterion

For general plasticity of metals, yielding is said to have occurred when the effective

stress reaches the tensile yield stress of the material obtained from a uniaxial tensile test. To

perform plasticity analyses when dealing with three-dimensional states of stress, von-Mises

effective stress is most often used in the yield criterion to calculate the effective stress when

dealing with ductile metals, see Equation 3-4.

( ) ( ) ( )[ ] 212

132

322

21

_

21 σσσσσσσ −+−+−= (Equation 3-4)

where, σ is the effective von Mises stress, 321 ,, σσσ are the principal stresses.

The Flow Rule

Constitutive stress-strain relationships that describe the path of plastic deformation of a

material are called flow rules. Flow rules for any yield criterion have the following form

(Mielnik, 1991):

( )( )λσσ

ε dF

dij

ijpij ∂

∂= (Equation 3-5)

where, ( )ijF σ is the isotropic yield function if the flow rule is considered associative,

pijε is the true plastic strain component, ijσ is the true stress component and λd is the plastic

compliance. The flow rule is termed an associated flow rule when the plastic potential surface

45

coincides with the yield surface. Metals conform to this rule quite well. For the von-Mises yield

criterion the flow rule obtained is:

λεεε

dS

dS

dS

d===

3

3

2

2

1

1 (Equation 3-6)

where, dλ is the plastic compliance and iS are the deviatoric stress components. The

deviatoric stress components are calculated as shown in Equations 3-7 and 3-8.

m

m

m

SSS

σσσσσσ

−=−=−=

33

22

11

(Equation 3-7)

where,

( )32131 σσσσ ++=m (Equation 3-8)

Deviatoric stress components are stresses that have the hydrostatic state of stress

subtracted out; this is done since plastic deformation is assumed to be a constant volume process

and thus hydrostatic stresses are assumed not to cause plastic deformation in the von Mises yield

criterion.

With,

σελ

23dd = (Equation 3-9)

where, _ε is the effective strain and

_σ is the effective stress.

Equation 3-5 can now be rewritten as follows:

46

( )

( )

( )⎥⎦⎤

⎢⎣⎡ +−=

⎥⎦⎤

⎢⎣⎡ +−=

⎥⎦⎤

⎢⎣⎡ +−=

213_

_

3

312_

_

2

321_

_

1

21

21

21

σσσσ

εε

σσσσ

εε

σσσσ

εε

dd

dd

dd

(Equation 3-10)

Note that since plastic deformation is assumed to be incompressible, Poisson’s ratio (ν), ν = 0.5.

The Hardening Rules

Hardening rules are used to describe the changing of the yield surface during progressive

yielding. These rules are implemented in finite element analysis codes to allow stress states for

subsequent yielding to be calculated. Generally, two hardening rules are available for metals,

isotropic hardening and kinematic hardening. When dealing with cyclic loads involving load

reversals, the Bauschinger effect should be observed. The Bauschinger effect states that the yield

strength of a metal will decrease when the direction of loading is changed.

In general, isotropic hardening works well for large deformations that occur without

cyclical loading. Figure 3-2 below shows how the isotropic hardening rule works.

Figure 3-2: Loading path generated by the isotropic hardening rule (ANSYS, 2004).

47

Note that the Bauschinger effect is not taken into account, so a material will have the

same yield strength in compression and tension during cyclic loading. This is unacceptable if the

analysis is concerned with cyclic loading, so isotropic hardening should only be used for

monotonic loadings.

Kinematic hardening is used when plastic deformations are small and loading involves

one or more reversals because it models the Bauschinger effect. As seen in Figure 3-3 below, the

yield stress is path dependent, that is the Bauschinger effect is taken into account. The yield

surface will not change in size, instead it translates as yielding progresses.

Figure 3-3: Loading path generated by the kinematic hardening rule (ANSYS, 2004).

It should be noted that in reality, metals exhibit a combination of isotropic and kinematic

hardening behaviour, but for simplicity one of the models is often used based on loading

conditions.

3.1.2 Creep

Creep deformation was already briefly introduced in Section 2.2.1. However, the

mathematical formulation of creep is presented in this section as part of the theory used to

complete this work. Figure 3-4 shows a typical creep curve measured during a creep rupture test.

48

Figure 3-4: Creep strain vs. time; typical plot generated from a creep rupture test (Kraus,

1980).

Examining Figure 3-4 it was initially determined that in order to model the creep process

in one dimension an expression of the following form was needed:

( )tTFc ,,σε = (Equation 3-11)

where cε is creep strain, σ is stress, T is temperature and t is time.

When developing mathematical creep relations it is customary to exclude tertiary creep

since it is often ignored during engineering creep analyses. This is because most engineering

analyses are not concerned with predicting failure. However, the mathematical development of

49

primary and secondary creep equations presented here is also applicable to tertiary creep. To

proceed with the development of mathematical creep relations the following creep expression is

assumed:

nmc tAσε = (Equation 3-12)

where, A, m and n are temperature dependent material constants.

The creep expression given by Equation 3-12 is known as the Bailey-Norton creep law.

A, m and n are material constants that are functions of temperature. This creep law models

primary and secondary creep. For variable stress problems, the creep strain rate is of interest.

Whether the creep strain rate is a function of time or of creep strain is an important factor and will

determine whether the creep law is termed time hardening or strain hardening. The time

derivative of the Bailey-Norton creep law is shown in Equation 3-13.

1−=∂∂

= nmc

c ntAt

σεε& (Equation 3-13)

where cε& is the creep strain rate.

Equation 3-13 shows the time hardening form of the Bailey-Norton creep law. In order

to derive the strain hardening form, first time needs to be solved for as a function of creep strain

(from Equation 3-12), see Equation 3-14.

n

m

c

At

1

⎟⎟⎠

⎞⎜⎜⎝

⎛=

σε

(Equation 3-14)

Substituting Equation 3-14 into Equation 3-13, the strain hardening form of the Bailey-

Norton creep law can be derived as shown in Equation 3-15.

50

( )( ) nncnm

nc nA /11 −= εσε& (Equation 3-15)

Figures 3-5 and 3-6 illustrate the difference between the time hardening and strain

hardening creep formulations.

Figure 3-5: Creep strain history prediction from time hardening theory (Kraus,

1980).

51

Figure 3-6: Creep strain history prediction from strain hardening theory (Kraus,

1980).

Under conditions of varying stress the time hardening and strain hardening solutions for

creep strain will differ. The time hardening law uses the creep strain rate at time, t, when the

stress changes to calculate a new creep strain rate. The strain hardening law uses the creep strain

rate at creep strain, cε , to calculate a new creep strain rate when the stress changes. From

52

experimental work, it has been shown, as expected, that the strain hardening law produces more

accurate results than the time hardening law under conditions of varying stress. However, creep

equations of the strain hardening form are typically more computationally intense to solve.

Creep deformation is most often analyzed in components under multiaxial stress

conditions. Therefore, it is important to derive multiaxial creep strain relations. Kraus (1980)

observed that the resulting multiaxial creep relations (assuming von Mises yielding behaviour for

ductile metals) must satisfy the following requirements:

• The multiaxial formulation must reduce to the uniaxial formulation when appropriate.

• The model must express the constancy of volume experimentally observed during creep

deformation.

• As observed experimentally, the model’s creep strain should not be influenced by the

hydrostatic state of stress.

• For an isotropic material, the directions of principal stress and strain should coincide.

Therefore, initial formulation of multiaxial creep strain was given by:

ijcij Sλε =& (Equation 3-16)

where, λ is the creep compliance and Sij is the deviatoric stress component.

The effective creep strain rate is given by Equation 3-16. Note that it is of the same form

as the von Mises effective strain which is used to combine the different strain components into a

single effective strain value.

53

( ) ( ) ( ) ( ) ( ) ( )[ ] 212

232

132

122

11332

33222

2211 632 cccccccccc εεεεεεεεεε &&&&&&&&&& +++−+−+−⎟⎟⎠

⎞⎜⎜⎝

⎛=

(Equation 3-17)

where cε& is the effective creep strain rate.

Now, if Equation 3-16 is substituted into Equation 3-17 and the equation of effective

stress is observed, Equation 3-4, it is found that:

cεσ

λ &23

= (Equation 3-18)

where, σ is the effective stress.

Substituting Equation 3-18 into 3-16, it is that found that:

cij

cij S ε

σε &&

23

= (Equation 3-19)

Therefore, to mathematically model the multiaxial creep strain rate, the effective creep

strain rate must be derived from uniaxial creep tests as follows. First, a uniaxial creep model is

developed to model the experimental creep curves. For example, an experiment that yields

primary and secondary creep curves could be modeled with the Bailey-Norton law, Equation 3-

20.


Converting strain and stress values to effective strain and stresses knowing that effective

quantities have been defined to reduce to the uniaxial forms when appropriate:


54

And the time derivative of Equation 3-21 is:

1−= nmc tAnσε& (Equation 3-22)

Therefore, for an assumed Bailey-Norton creep law, the time hardening multiaxial creep

equation is given by Equation 3-23.

1

23 −= nm

ijcij tAnS σ

σε& (Equation 3-23)

Which can be further simplified to Equation 3-24.

11

23 −−= nm

ijcij tAnS σε& (Equation 3-24)

The strain hardening multiaxial creep equation (Equation 3-25) is obtained by

substituting the value of t (from Equation 3-21) into Equation 3-24.

( ) ( ) nn

cnm

nij

cij nAS

111

23 −

−= εσε& (Equation 3-25)

This is how one would mathematically model multiaxial creep using uniaxial creep

rupture data assuming von Mises yielding behaviour.

3.1.3 Creep Damage Equations

From the literature review conducted in Section 2.2, it was determined that the Kachanov

One-State Variable technique and the MPC Omega method were most suitable to predict stress

rupture of pressure vessels exposed to accidental fires. The equations needed to model creep and

creep damage using these techniques are presented Sections 3.1.3.1 and 3.1.3.2.

55

3.1.3.1 Kachanov One-State Variable Technique

The original Kachanov/Rabotnov creep damage law, derived empirically, is:

( )

r

A ⎥⎦

⎤⎢⎣

⎡−

=ω

σω1

& (Equation 3-26)

where, ω& is creep damage rate, σ is true stress, A and r are material constants and ω is

creep damage.

Equation 3-27 shows how creep damage was coupled with creep strain to model tertiary

creep.

nc A ⎟

⎠⎞

⎜⎝⎛−

=ω

σε1

'& (Equation 3-27)

where, cε& is the creep strain rate, A’ and n are material constants, σ is stress and ω is

creep damage.

Both equations were derived empirically to mathematically model experimentally

measured tertiary creep behaviour. Equation 3-26 describes how creep damage will increase and

Equation 3-27 describes how creep strains will increase.

The One-State Variable equations as applied by Hyde et al. (2006) have more constants

added to them in an effort to more accurately predict tertiary creep behaviour.

The uniaxial forms of the One-State Variable equations are:

mn

c tA ⎟⎠⎞

⎜⎝⎛−

=ω

σε1

'& (Equation 3-28)

( )mtB φ

χ

ωσω−

=1

'& (Equation 3-29)

where, t is time and B’, φχ , and m are material constants.

56

The multiaxial forms of the One-State Variable equations, Equations 3-30, 3-31 and 3-32,

are derived using the creep theory presented in Section 3.1.2.

mijn

cij t

SA

σωσε ⎟

⎠⎞

⎜⎝⎛−

=1

'23

& (Equation 3-30)

( )mr tB φ

χ

ωσ

ω−

=1

'& (Equation 3-31)

( )σαασσ −+= 11r (Equation 3-32)

where, cijε& is the multiaxial creep strain rate component, σ is the effective stress, Sij is the

deviatoric stress component, rσ is the rupture stress, 1σ is the maximum principal stress and α is

a material constant related to multiaxial stress-state behaviour.

Note that creep constant n, in Equations 3-28 and 3-30, does not necessarily have to be equal to

the Norton creep constant because the values of these creep constants are often optimized to best

fit experimental uniaxial creep rupture data.

3.1.3.2 MPC Omega Method

The MPC Omega Method equations were developed empirically to model the

microstructural mechanisms that govern tertiary creep in order to match experimentally measured

tertiary creep behaviour.

The exponential approximation shown in Equation 3-33 was found to hold for levels of

20% true creep strain or more on the strain-time curve (Prager, 1995).

ccc cpmo

c eee εεεεε && = (Equation 3-33)

57

where, cε& is the creep strain rate, 0ε& is the initial creep strain rate (material constant), cε

is the creep strain, m is Norton’s exponent to account for the rate increase due to cross section

reduction (stress increase), p corresponds to micro-structural damage, and c is used to account for

deficiencies in Norton’s exponent and other micro-structural factors associated with the stress

change.

Note that Equation 3-33 can be re-written as:

ccpmo

c e εεε )( ++= && (Equation 3-34)

The exponents in Equation 3-34 are then gathered as follows:

pcpm Ω=++ (Equation 3-35)

cpeo

c εεε Ω= && (Equation 3-36)

where, pΩ is Omega (material constant).

The multiaxial creep strain rate, Equation 3-37, was derived using the creep theory presented in

Section 3.1.2.

cpe

Sijcij

εεσ

ε Ω⋅⋅= 023

&& (Equation 3-37)

where, cijε& is the multiaxial creep strain component, Sij is the deviatoric stress component,

σ is the effective stress and cε is the effective creep strain.

And Ωp can be found from experimental data in the following way (see Figure 2-7):

pc

c

dd

Ω=εε&ln

(Equation 3-38)

A useful equation, that was derived empirically, for determining the life fraction consumed is:

58

1+Ω

Ω==

psc

psc

r

s

tt

ConsumedFractionLifett

εε&

& (Equation 3-39)

where, ts is the time in service, tr is the rupture time, cε& is the effective creep strain rate.

To extend the MPC Omega method to multiaxial loading and tube loading the Omega creep

constant needs to be modified. For example, for tubes Ωp becomes (Prager, 2000):

nuniaxialtube +Ω=Ω (Equation 3-40)

where, n is the Norton creep exponent.

3.2 FEA Governing Equations

The following section summarizes the main governing equations used by the finite

element analysis solver.

The pressure vessel’s numerical finite element analyses will model the structure’s elastic,

plastic and creep deformations where necessary.

FEA code is based on solving numerous finite elements (mesh) which discretize the

domain to achieve a converged solution which is in equilibrium. To achieve the solution, several

conservation equations (i.e. mass, momentum and energy) are solved. The governing equations

listed below are used by FEA codes to numerically model the physical phenomena of structural

deformation.

Mass

( ) 0=+∂∂ vdiv

tρρ

(Equation 3-41)

59

Linear Momentum

vbdiv &ρσ =+ (Equation 3-42)

Angular Momentum

Tσσ = (Equation 3-43)

Mechanical Energy

∫∫∫∫ΩΩ∂ΩΩ

⋅+⋅=+ dvvbdsvtdvddvvDtD :

21 2 σρ (Equation 3-44)

Virtual Work

( ) ∫∫Ω∂Ω

=⋅−⋅− 0: dsutdvube δδδσ (Equation 3-45)

These governing equations are used to solve the global system matrix, Equation 3-46

FuKuCuM =++ &&& (Equation 3-46)

where, M is the mass matrix, C is the damping matrix, F is the load vector and u is the

unknown displacement vector.

For the case of a quasi-static analysis, Equation 3-46 can be simplified to Equation 3-47.

FuK = (Equation 3-47)

3.3 FEA User Subroutines

Commercial FEA computer codes allow the user to implement user subroutines in order

to generate customized versions of the computer codes. The user subroutines are most frequently

written in the Fortran computer programming language. Common applications of user

60

subroutines include creating custom material models, elements and loadings. Most commercial

FEA computer codes offer the option of implementing about 40 different user subroutines.

ABAQUS (2007) provides the flow chart shown in Figure 3-7 to show where common user

subroutines fit into the analysis process.

CREEP, as shown in Figure 3-7, is a user subroutine that allows the user to implement

custom creep equations. DLOAD allows the user to implement custom loadings. For more

information on various subroutines offered by ABAQUS see (ABAQUS, 2007).

For this work, creep user subroutines were used to implement the Kachanov One-State

Variable and MPC Omega creep and creep damage equations into ANSYS and ABAQUS. The

UTEMP user subroutine was used in ABAQUS to define temperatures as loads at nodes.

61

Figure 3-7 Where various user subroutines fit into FEA (ABAQUS, 2007)

62

Chapter 4 SA 455 Mechanical Testing

Experimental creep rupture and elevated temperature uniaxial tensile tests were

preformed to measure needed mechanical properties of SA 455 steel. These properties were

previously unavailable because SA 455 steel is not meant for high temperature applications.

4.1 Experimental Specimens

Tensile specimens were cut with a band saw from a plate 7.1 mm thick, 2.11 m wide and

3 m long. The chemical composition and tensile properties reported by the supplier were

compared to the ASTM requirements in Tables 4-1 and 4-2. The comparison showed that the SA

455 samples conformed to the ASTM requirements of SA 455 steel. Specimens conforming to

ASTM E 21-92 and E 139 -96 specifications were produced, as discussed in Section 2.1. Figure

4-1 shows the dimensions of the rectangular cross sectioned specimens. The reduced area in the

centre of the specimen is the gage length. The errors presented in Table 4-2 are measurement

errors.

Table 4-1: Chemical composition of SA 455.

Chemical

composition (wt.%)

Supplier

analysis

ASTM SA 455

requirement [3]

Carbon 0.26 0.33 max

Manganese 1.14 0.79 – 1.3

Phosphorus 0.013 0.035 max

Sulfur 0.006 0.035 max

Silicon 0.3 0.45 max

Iron rem. rem.

63

Table 4-2: Tensile properties of SA 455

Tensile properties Supplier

specified

ASTM

SA 455

requirement

Present Study*

Yield Strength, MPa 420 260 min 423 ± 0.41%

Tensile Strength, MPa 610 515-655 628 ± 0.41%

Elongation, % 21.8A 22B 23.9C ± 0.25%

*Performed at Queen’s University, Kingston, Ontario. AOver a 1.66” gauge length. BOver a 2” gauge length. COver a 1.5” gauge length.

Figure 4-1: Specimen dimensions, the reduced area in the centre is the gauge length;

specimen thickness is 7.1, all dimensions are in mm.

4.2 Experimental Apparatus

The elevated temperature tensile and creep rupture tests were performed with an 8500

series Instron servo-hydraulic testing instrument in the Department of Mechanical and Materials

Engineering at Queen’s University. The test rig is shown in Figure 4-2. A 100 kN dynamic load

cell with an error of ±0.25% (Instron, 2008) was used. Since a high temperature extensometer

64

was not available, displacement, later converted to strain, was measured by the Instron’s linear

variable differential transformer (LVDT) with an error of ±0.25%. The error of the LVDT was

assumed to be the same as the error of the load cell since no data was found to indicate otherwise.

Figure 4-2: Instron with furnace, custom grips and sample.

Custom grips, shown in Figures 4-3 and 4-4, were designed and manufactured to hold the

test specimens in the furnace. The grips were machined from super alloy A-286, solution treated

(982oC) and aged. A-286 has excellent high temperature strength and corrosion resistance, and

guaranteed that the deformation within the grips during the creep-rupture testing was negligible

compared to that in the gauge length of the specimen. The upper custom grip was attached to a

universal coupling that has 2 degrees of rotational freedom, thus minimizing bending forces

applied to the samples. Insulation rings show in Figure 4-4 were used to minimize heat transfer

from the specimen to the grips.

65

Figure 4-3: Custom grip’s design drawing.

Figure 4-4: Custom tensile test grip.

66

A custom made radiant heat transfer clamshell furnace, shown in Figure 4-5, that mounts

onto the Instron machine was used to heat the samples.

Figure 4-5: Custom clamshell furnace.

The heater’s maximum operating temperature was 980oC and its power output was

1130W at 120 VAC. Special ceramic fibre insulation was used to plug the top and bottom of the

furnace openings to minimize heat loss from the furnace to the surroundings. An Omega

CN2100-R20 temperature controller, shown in Figure 4-6, was used to control the sample

temperature.

Figure 4-6: Omega CN2100-R20 temperature controller.

67

4.3 Data Acquisition

Each sample had 3 type K thermocouples spot welded onto it; one at the centre of the

gage length and the other 2 at the top and bottom ends of the gage length. The type K

thermocouple has a standard error of ± 2.2oC (Omega, 2008). The thermocouple at the centre of

the gage length was used by the Omega controller to control the furnace. The thermocouples

were spot welded to the gage length with wire ends welded onto opposite sides of the specimen.

This was done such that the sample physically became the thermocouple bead and thus error due

to bead contact to the sample and surrounding air could be eliminated.

Thermocouple preparation included:

• Spot welding each thermocouple to the gage length.

• Ensuring that there was no wire contact behind the weld. This was done to prevent shorting of

the thermocouple from taking place.

• Ceramic insulators were strung over the thermocouple wires in the hot zone and Teflon

covering was used to insulate the remaining wire.

An IOtech Personal DaqView 55, personal data acquisition system, was used to collect

and store temperature data from the thermocouples at the ends of the specimen’s gage length, see

Figures 4-7, 4-8 and 4-9. The thermocouple placed at the centre of the gage length was used by

the Omega temperature controller to control the furnace.

68

Figure 4-7: Data acquisition system used to collect specimen temperatures from the top and bottom of the gage length.

0

100

200

300

400

500

600

700

0:00:00 0:07:12 0:14:24 0:21:36 0:28:48 0:36:00 0:43:12 0:50:24 0:57:36 1:04:48 1:12:00

Time

Tem

pera

ture

(o C)

Top of Gauge LengthBottom of Gauge Length

Figure 4-8: Temperature history of top and bottom of gauge length during a creep test at 630oC; start up heating, 20 minute hold time and creep test time are shown.

69

625

626

627

628

629

630

631

632

633

634

0:40:19 0:43:12 0:46:05 0:48:58 0:51:50 0:54:43 0:57:36 1:00:29 1:03:22 1:06:14

Time

Tem

pera

ture

(o C)

Top of Gauge LengthBottom of Gauge Length

Figure 4-9: Temperature history of top and bottom of gauge length during a creep test at 630oC; 20 minute hold time and 5.7 minute test are shown.

Displacement and load were read and stored by a computer connected to the Instron’s

computer system. These measurements were made once every second during the creep rupture

tests and four times a second during the tensile tests. A high temperature extensometer was not

available during testing so the displacements were read by the Instron’s LVDT. The stiffness of

our system at various temperatures was later determined (see Section 4.5) so that all elastic strains

occurring inside and outside the specimen’s gage length could be removed from the data.

4.4 Experimental Details

The specimens took about 50-70 minutes to reach testing temperature and then were held

at the test temperature for 20 minutes for the elevated temperature tensile and creep rupture tests.

The ASTM standard calls for an hour of holding time for creep rupture tests; however, these

creep rupture tests are performed at high enough temperatures and stresses to produce failure in

under 100 minutes. Therefore, it was decided that the 20 minute holding time indicated for

elevated temperature tensile tests would be more appropriate. Due to the unusually long tensile

70

samples much difficulty was experienced when attempting to hold the temperature of the gauge

length to ±2oC during the tests. Due to this difficulty, the test temperature tolerance was relaxed

to +2oC; -4oC, see Figure 4-9. Near the end of the tests, the temperatures may have risen as high

+4oC as instability and necking caused the gage length to move significantly in the furnace. This

is acceptable according to the creep rupture testing standard provided by ASME (ASME E139-

96, 1998) since the samples were not overheated (except slightly at the end of the tests) and

exposure time at low temperatures was not significant; therefore, the test results were considered

accurate. Otherwise, the creep rupture tests conformed to ASTM E 139-96 standards.

The load was ramped such that it took 5-10% of the total test time to reach the desired

test load. The creep rupture tests were conducted at 550oC, 600oC, 630oC, 660oC, 690oC and

720oC. The test loads were chosen such that the creep rupture tests varied in failure time from 3

to 100 minutes. These are extremely low failure times as far as creep-rupture is concerned.

However, this is the failure window of the studied application, so these were the conditions

tested.

The elevated temperature tensile tests conformed with ASTM E 21-92 standards. The

elevated temperature tensile tests were performed at 400oC, 500oC, 600oC and 720oC. No failure

was observed to occur at the location of any spot weld, thus it was assumed that the results were

not affected by the thermocouple welds, see Figure 4-10.

71

Figure 4-10: Failed specimen with 3 thermocouples spot welded onto it.

4.5 Elevated Temperature Tensile Tests

Room and elevated temperature tensile tests were performed on SA 455 steel. An

extensometer was only used for the room temperature testing. At high temperatures, the

specimen elongation was measured from the Instron’s linear variable differential transformer

(LVDT). The measured displacement included the elastic deformation in the non-gage length

section of the sample, the grips, the load train and the elastic and plastic deformation in the gauge

length of the specimens. The stiffness of the test rig and specimen was measured at each test

temperature by loading-unloading experiments. Using these stiffness values, all the elastic

deformations were subtracted out of the total deformation data. Consequently, a data set of only

72

the specimen’s plastic deformation was obtained. Table 4-3 shows the temperature dependency

of SA 455’s yield stress and ultimate tensile strength found in the elevated temperature tensile

tests. The temperature dependency of Young’s modulus of elasticity as reported by ASME (2004)

for medium carbon steels is also shown in Table 4-3 for reference.

To verify that the measured values of SA 455 are valid, they were compared to published

values of TC 128 (Birk et al., 2006) and a plain medium carbon steel (ASME, 2004), see Figure

4-11. The comparison showed, as expected, that TC 128 was stronger at higher temperatures and

that plain medium carbon steel was significantly weaker at all temperatures.

Table 4-3: Measured SA 455 temperature dependency of yield stress and UTS.

Temperature ± 2.2oC [oC]

Yield Stress ± 0.41% [MPa]

UTS ± 0.41% [MPa]

E (ASME,2004) [GPa]

22 423 628 201 400 415 617 169 500 285 420 148 600 195 225 121 720 92 92 83

73

0

100

200

300

400

500

600

700

0 200 400 600 800 1000

Temperature [oC]

Ulti

mat

e Te

nsile

Str

engt

h [M

Pa]

SA 455

TC 128

MediumCarbonSteel

Figure 4-11: Comparison of ultimate tensile strength of SA 455, TC 128 and medium carbon steel.

Figure 4-12 shows a plot of the temperature dependency of the true stress versus true plastic

strain curves. At room temperature (22°C) there was observed a detectable yield point and Lüders strain,

which disappears at higher temperatures. The Lüders strain is commonly observed in mild and medium

carbon steels, it is the region of yielding that displays almost perfectly plastic behaviour as plastic

deformation begins due to unpinning of dislocations from nitrogen and carbon atmospheres (Hall, 1970).

Higher temperatures are characterized by lower yield strengths and reduced work hardening. This data is

needed to conduct high temperature elastic-plastic finite element analyses of SA 455 steel.

74

0

100

200

300

400

500

600

700

800

0 0.05 0.1 0.15 0.2

True Plastic Strain

True

Stre

ss [M

Pa]

22°C400°C500°C600°C720°C

Figure 4-12: SA 455 steel true stress true strain plot at various temperatures.

From Figure 4-12 it is shown that at 720oC, SA 455 steel experiences little plastic

hardening and it quite weak. This can be explained by the fact that at 727oC steel’s

microstructure changes from ferrite to austenite, a much weaker configuration (Callister, 1997).

4.6 Creep Rupture Tests

The creep rupture experimental results are summarized in Table 4-4. Note that since a

high temperature extensometer was not used, the specimen extension was determined from the

Instron’s actuator LVDT, which includes elastic deformation in the entire specimen, grips and

load train. The LVDT displacement was converted to specimen extension by dividing it by the

sample’s gauge length. Later on, all the elastic strains were subtracted out of the data using the

system stiffness that was measured at all temperatures. The N/As in Table 4-4 represent tests for

which computer problems caused no test data to be stored.

75

Table 4-4: Stress-rupture properties of SA 455 steel

Mean temperature ± 2.2oC

[oC]

Initial stress ± 0.41% [MPa]

Rupture time [min]

Elongation* at rupture ± 0.25%

[%] LVDT sensor

Elongation** at rupture ± 0.05%

[%] post measurement 550 326.7 1.90 30 30.7 550 296.3 7.57 33.5 32.7 550 266 21.57 36.3 37.5 550 265.8 22.81 35.3 34.0 600 220.5 4.84 N/A 41.2 600 196.3 10.00 45.1 44.8 600 177.3 20.69 48.4 47.2 600 151.3 67.43 49.9 50.6 630 176.7 5.72 N/A 50.3 630 175 5.62 54.9 55.4 630 149.3 15.29 52.9 54.0 630 130 47.56 56.2 56.6 660 150 3.56 50.8 51.2 660 125 12.78 57.9 60.7 660 106.7 29.26 N/A 51.7 690 113 4.30 56.8 61.3 690 100 10.53 64.9 65.9 690 77 38.33 67.2 64.2 720 89.3 3.45 72.4 71.1 720 74 13.27 76.8 76.4 720 50 80.57 107.8 105.4

*Based on a 1.5” gauge length. Deflection measurements were made by the Instron, no extensometer

used.

**Based on a 1.5” gauge length. Measured with vernier calipers.

The test specimen gauge length was measured after fracture with vernier calipers, and

compared to the Instron reported value at rupture.

SA 455 steel’s stress rupture data presented in Table 4-4 is plotted in Figure 4-13 as stress

versus time to rupture. For comparison purposes TC 128’s stress rupture is plotted in Figure 4-14; TC

128 is a pressure vessel steel that has similar composition to SA 455 steel. The failure times for SA

455 steel and TC 128 steel are similar as expected.

All SA 455 steel creep rupture tests produced conventional creep curves. The creep curves

were all similar in that the tertiary stage of creep is the most prominent due to the relatively quick

creep failure times. Examples of these plots from various testing conditions are shown in Figure 4-15;

all the creep curves are shown in Appendix A. Note that the elastic strains in the sample, load train

76

and grips were subtracted out of the total strain to give creep strain. This was accomplished by

measuring the stiffness of our test rig (specimen, load train and grips) at the various temperatures.

Once this stiffness was found, all elastic strains were removed from the data.

0

50

100

150

200

250

300

350

1.00 10.00 100.00

Rupture Time [min]

Stre

ss [M

Pa]

550°C600°C630°C660°C690°C720°C

Figure 4-13: Plot of SA 455 steel’s stress rupture data.

Figure 4-14: Plot of TC 128 steel’s stress rupture data (Birk et al., 2006).

77

00.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0 1000 2000 3000 4000 5000

Time (s)

True

Cre

ep S

train

S=151.3 MPaS=177.3 MPaS=196.3 MPa

Figure 4-15: Measured SA 455 steel creep curves at 600oC.

Steady state creep of a material is most often modeled using Norton’s creep relationship given

in Equation 3-20. Note that the stress and strain in Norton’s creep relationship are either true or

engineering stress and strain, based on if the creep rupture data is from a constant true or engineering

stress test. In order to derive Norton’s constants, one needs minimum creep rates under various

conditions. Table 4-5 provides the minimum creep rates of SA 455 steel for our various testing

conditions. In the creep rupture tests performed there was no significant primary or secondary creep,

therefore, a minimum creep rate was based on the average of the creep rates during the first 30% of the

test duration.

Due to the high loads at which the creep rupture tests were performed, creep failure times are

relatively short and the shape of the creep curve is dominated by tertiary creep behaviour. However,

even under these conditions, minimum creep rates as well as Norton’s creep constants can be useful.

78

Table 4-5: Minimum creep rates of SA 455 steel.

Mean temperature ± 2.2oC

[oC]


Minimum creep rate ± 0.5%

[strain per second] 550 326.7 1.10E-03 550 296.3 2.40E-04 550 266 7.08E-05 550 265.8 5.74E-05 600 196.3 1.95E-04 600 177.3 9.35E-05 600 151.3 1.55E-05 630 176.7 4.08E-04 630 149.3 1.23E-04 630 130 2.83E-05 660 150 6.40E-04 660 125 1.59E-04 660 106.7 3.23E-05 690 113 6.37E-04 690 100 1.69E-04 690 77 3.30E-05 720 89.3 9.30E-04 720 74 1.61E-04 720 50 1.85E-05

4.7 Creep Data Correlation

A correlation through all temperatures for the creep rupture data was desired. Given a

temperature and stress the correlation accurately calculated failure times. When correlation failure

times diverge from experimental failure times it was important that the correlation values remain

conservative since a conservative analytical model was desired.

Larson-Miller (1952) is a widely used and accepted correlation of stress rupture data. The

Larson-Miller model correlates stress rupture data using the Larson-Miller parameter (P), which is

defined as follows:

PL-M = T(CL-M + log tr) (Equation 4-1)

79

where, PL-M is the Larson-Miller parameter, T is temperature measured in oR and tr is rupture

time measured in hours. The Larson-Miller constant was originally indicated as taking on a value of

20 (Larson and Miller, 1952).

Since the creep rupture testing was conducted under conditions that generated unusually fast

creep failures the validity of using CL-M=20 was investigated. If one plots log of failure time versus

1/T and draws lines of constant stress, the y-intercept of the lines should be roughly the same value, -

CL-M. Such a plot for the measured SA 455 data is shown in Figure 4-16. Due to the testing conditions

only 3 lines of constant stress could be drawn; the tests were conducted at constant temperature not

constant stress conditions.

After examining Figure 4-16, it was apparent that a Larson-Miller constant between 17 and 20

would best suit the data set. It was found that a value of CL-M=19 gave the most accurate correlation

while being conservative. The Larson-Miller plot, with CL-M=19, is presented in Figure 4-17. The

values of the Larson-Miller parameter, PL-M, are listed in Table 4-6.

-25

-20

-15

-10

-5

0

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007

1/T [oR-1]

log(

Failu

re T

ime[

hr])

S=175 MPaS=150 MPaS=130 MPa

Figure 4-16: Investigating the suitability of CL-M=20 for SA 455 steel.

80

0

50

100

150

200

250

300

350

400

24000 26000 28000 30000 32000 34000 36000

Larson-Miller Parameter PL-M

Stre

ss [M

Pa]

Figure 4-17: Larson-Miller correlation for SA 455 steel, CL-M=19.

Table 4-6: Larson-Miller parameter values

Mean temperature ± 0.75% [oR]


Larson-Miller parameter

1481.7 326.7 25930.1 1481.7 296.3 27493.3 1481.7 266 27529.5 1481.7 265.8 26819.6 1571.7 220.5 29134.9 1571.7 196.3 29941.5 1571.7 177.3 28638.7 1571.7 151.3 28142.8 1625.7 176.7 29228.9 1625.7 175 29215.4 1625.7 149.3 29922.3 1625.7 130 30723.7 1679.7 150 29853.6 1679.7 125 30785.4 1679.7 106.7 31389.9 1733.7 113 31629.8 1733.7 100 32602.4 1733.7 77 30955.8 1787.7 89.3 32794.1 1787.7 74 34194.6 1787.7 50 31749.4

81

To show the accuracy of this correlation, Figure 4-18 plots the correlation predicted failure

times on the vertical axis and experimental failure times on the horizontal axis. The line y=x

represents when the two times are exactly equal. Points below the y=x line are conservative

predictions made by the correlation.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.50 1.00 1.50

Experimental Failure Time [hr]

Cor

rela

tion

Failu

re T

ime

[hr]

Figure 4-18: Correlated vs measured failure times.

The equation of the second order polynomial used to fit the data in Figure 4-17 is as follows:

3126 10742.3)10042.2()10813.2( ×+×−×= −−

−−

MLML PPσ (Equation 4-2)

For many applications it is useful to have an equation that would generate failure times as a

function of temperature and stress. The first step to such an equation would be to solve the above

equation for PL-M, the Larson-Miller parameter.

)1081.2(2

)1074.3)(1081.2(4)1004.2(1004.26

36211

−

−−−

− ×−××−×−×

=σ

MLP (Equation 4-3)

82

Note that stress in Equation 4-3 was measured in MPa. It should also be noted that the ± is

simply a subtraction, since the addition solution is physically meaningless. The solution for PL-M can

be entered into Equation 4-4 to produce the failure time as follows.

19)/(10 −−= TP

rMLt (Equation 4-4)

where, tr is rupture time in hours.

Note that generally the Larson Miller correlation has been shown to provide good predictions

for extrapolated creep rupture data (Kraus, 1980). However, the creep rupture tests performed in this

study were at relatively high stresses, so rupture times were low. Therefore, caution and engineering

judgment is recommended when using this correlation to extrapolate creep failure times beyond 100

minutes.

83

Chapter 5 SA 455 Steel Creep and Creep Damage Constants

From the literature review summarized in Section 2.0 the Kachanov One-State Variable and

MPC Omega method were the creep damage models selected to numerically predict the tertiary creep

behaviour of SA 455 steel. The creep constants for these models were extracted from the experimental

creep rupture tests. This section presents the values of the creep constants and outlines the procedure

used to compute them from experimental data.

5.1 Kachanov One-State Variable Constants

The method used to determine the temperature dependent material constants m, n, φ , χ, B’

and A’ was described in great detail by Hyde, Sun and Tang (1998).

With some simple equation rearrangements, the creep strain was normalized by that of the

failure creep strain to give:

( )

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

+−+ )1/(11

11

φ

εε

nm

rcf

c

tt

(Equation 5-1)

where cε is creep strain, cfε is creep strain at rupture, t is time, tr is rupture time and m, n, and

φ are material constants.

In order to determine m,n andφ material constants, a three variable least squares optimization

approach was conducted. The least squares optimization method is given by:

( )2

1),,,(∑

=

−=n

iii nmtfyF φ (Equation 5-2)

84

where, F is to be minimized, yi is the normalized experimental data and ),,,( φnmtf i is the

calculated normalized creep strain based on Equation 5-1.

This process gives the optimized values of m, n andφ and will produce the best fit to the

normalized experimental data plots. Once material constants m, n andφ are found, χ and B’ can be

found by plotting the natural logarithm of failure time versus the natural logarithm of nominal test

stress. Figure 5-1 shows this plot for SA 455 steel.

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

3.75 4.25 4.75 5.25 5.75 6.25

ln (Initial Stress [MPa])

ln(R

uptu

re T

ime

[s]) 550°C

600°C630°C660°C690°C720°C

Figure 5-1: Relationship between natural logarithm of rupture time and natural logarithm of

initial stress; plot needed to compute χ and B’ for SA 455.

The material constant χ is given by the negative of the slope of the linear trend-line that fits

the data and B’(φ +1) is given by exp( - yintercept). Therefore, the only material constant left to

determine is A’. The value of A’ can be determined by using Equation 5-3. Equation 5-3 was derived

85

by integrating the Kachanov One-State Variable creep equation by time and isolating the strain at time

of failure. Details of this process are presented by Hyde et al. (1998).

( ) ( )[ ])1/(11' )1(

+−+=

+

φσ

εnmtA m

fn

cf (Equation 5-3)

The material constants for SA 455, found as discussed above, are summarized in Table 5-1.

Table 5-1: One-state variable continuum damage mechanics material constants for SA 455.

Temperature (oC) A’ n m B’ φ χ 550 1.1x10-17 5.3 0 3.8x10-33 6.5 11.7 600 9.1x10-18 5.7 0 1.6x10-20 6.7 7.0 630 2.1x10-17 5.8 0 1.2x10-19 6.4 6.9 660 2.3x10-16 5.6 0 1.9x10-17 6.6 6.2 690 5.9x10-16 5.8 0 1.7x10-15 6.5 5.6 720 3.5x10-15 5.8 0 2.5x10-14 6.5 5.3

The temperature dependency of material constants n, φ , χ, B’ and A’ was plotted in Figures

5-2 and 5-3. The zero values for m are not concerning since the value of m is often 0 or close to 0 for

most steels (Hyde et al., 2006 and 1998). The values of the natural logarithms of A’ and B’ increased

linearly with temperature. The exception to this was the value of ln(B’ ) at 550oC. The values of φ

showed little dependence on temperature. The values of n increased linearly with temperature and x

decreased exponentially with temperature. In the literature review, no work was found that addressed

the temperature dependencies of these creep constants. Therefore, the author cannot compare these

trends to trends observed by others.

86

-80-75-70-65-60-55-50-45-40-35-30

500 600 700 800

Temperature (oC)

Mat

eria

l Con

stan

t Val

ue

ln B'ln A'

Figure 5-2: Temperature dependence of A’ and B’.

5

6

7

8

9

10

11

12

500 550 600 650 700 750

Temperature (oC)

Val

ue o

f Con

stan

t

n

Ø

x

Figure 5-3: Temperature dependence of n, φ , χ

5.2 MPC Omega Constants

The two MPC Omega material constants, Ωp and 0ε& , are dependent on temperature and stress.

Omega (Ωp) is the slope of the line that fits the plot of the natural logarithm of the true creep strain rate

versus the true creep strain. This can be seen by the following equation:

87

pc

c

dd

Ω=εε&ln

(Equation 5-4)

where, cε& is the creep strain rate, cε is the creep strain and pΩ is the material constant omega.

An example of such a plot used to determine Ωp is shown in Figure 5-4.

Once Ωp is known, the initial creep strain rate constant ( 0ε& ) is found using Equation 2-4 and

the experimental data. Norton’s creep exponent, n, was found using the data from Table 4-5. Table 5-

2 presents the Omega MPC material constants for SA 455. The stress dependency of Ωp and 0ε& are

plotted in Figures 5-5 and 5-6.

y = 11.866x - 9.57

-12

-10

-8

-6

-4

-2

00 0.1 0.2 0.3 0.4

True Creep Strain

Nat

ural

Log

of T

rue

Cre

epS

train

Rat

e

Figure 5-4: Determining SA 455 steel’s value of Omega for a test temperature of 600oC and

stress of 177.3 MPa.

88

Table 5-2: SA 455 steel Omega MPC creep constants

Temperature (oC) Stress (MPa) Ωp 0ε& n

550 326.7 12 6.80x10-4 13.66 550 296.3 15.8 1.36x10-4 13.66 550 266 20.5 3.77x10-5 13.66 600 196.3 10 1.62x10-4 9.87 600 177.3 11.9 6.72x10-5 9.87 600 151.3 14.5 1.70x10-5 9.87 630 176.7 9 3.14x10-4 8.64 630 149.3 11 9.82x10-5 8.64 630 130 13 2.69x10-5 8.64 660 150 8 5.57x10-4 8.74 660 125 10 1.29x10-4 8.74 660 106.7 11.5 4.93x10-5 8.74 690 113 9.4 4.18x10-4 7.49 690 100 10.1 1.56x10-4 7.49 690 77 11.6 3.74x10-5 7.49 720 89.3 9 5.38x10-4 6.57 720 74 9.2 1.36x10-4 6.57 720 50 9.6 2.17x10-5 6.57

4

6

8

10

12

14

16

18

20

22

0 100 200 300 400

Initial Stress [MPa]

Om

ega,

Ωp

550°C600°C630°C660°C690°C720°C

Figure 5-5: Stress dependency of Ωp.

The values of Ωp decreased linearly with stress and decreased with temperature. The values of

0ε& increased exponentially with stress and increased with temperature. These trends could not be

89

directly compared to the trends of the temperature and stress dependencies of Ωp and 0ε& as given by

the API Fitness for Service because of the vast difference in failure time predictions. However, the

general trends do compare well with those observed by Prager (2000).

0.E+00

1.E-04

2.E-04

3.E-04

4.E-04

5.E-04

6.E-04

7.E-04

8.E-04

0 50 100 150 200 250 300 350

Stress [MPa]

Initi

al T

rue

Cre

ep S

train

Rat

e [s

-1]

550°C600°C630°C660°C690°C720°C

Figure 5-6: Stress dependency of 0ε& .

90

Chapter 6 Numerical Analysis of Tensile Specimens

The creep and creep damage Equations 3-28, 3-29 and 3-34, 3-39 were implemented into both

ANSYS and ABAQUS via usercreep subroutines using the values of the creep constants found in

Section 5.0. Creep damage was calculated as a solution dependent variable in the user subroutines.

The gage length of the test specimens used in the lab was modeled for the numerical analysis work.

Failure was determined to have occurred when the damage parameter or life fraction consumed was

equal to 1 in elements throughout the thickness of the samples or when the analyses became

numerically unstable due to high creep strain rates. The objective of these finite element analyses was

to validate the creep damage models for the case of uniaxial creep loading as well as to assess the non-

linear creep analysis capabilities of both ANSYS and ABAQUS.

6.1 Tensile Specimen Model

In order to model the structural high temperature failure of pressure vessels, a material model

capable of predicting the high-temperature creep behaviour of SA 455 steel was required.

Since creep of steels is usually studied on the time scale of hundreds or thousands of hours, it

was not clear which creep damage model(s) would best suit this application.

The objective of Section 6 was to examine if the chosen creep damage models would

accurately predict the high-temperature creep rupture behaviour of SA 455 steel and to validate the

creep constants found in Section 5. From the literature review summarized in Section 2, it was

determined that the Kachanov One-State Variable Technique and MPC Omega Method would be best

suited to model the high-temperature creep behaviour of SA 455 pressure vessel steel.

Commercial finite element analysis computer codes ANSYS and ABAQUS are both used in

order to examine their performance when modeling relatively fast tertiary creep. An in-depth

91

comparison between experimental uniaxial creep rupture properties and those predicted by the finite

element analyses was conducted.

6.1.1 Analysis Geometry and Mesh

The dimensions of the tensile specimen being modeled are shown in Figure 4-1. The

objective of the numerical analyses was to predict the experimentally measured creep behaviour of the

tensile specimen’s gage length. Therefore, only the gage length of the specimen was modeled. Figure

6-1 shows the part model used in the numerical analyses. From Figure 6-1, it is clear that the model

includes a bit of specimen length outside of the gauge length at the top and bottom of the model.

These lengths were included so that boundary conditions and loading do not have any local effects on

the gage length; this was later confirmed by the analyses. The boundary conditions and loads are

shown in Figure 6-2.

Figure 6-1: Numerical model of the tensile specimen’s gage length.

92

Figure 6-2: Numerical analysis boundary conditions and loads, bottom face is fixed in all

directions, pressure load applied on top face.

In ABAQUS, 448 C3D8R elements were used to mesh the gauge length model, see Figure 6-

3. C3D8R is a three-dimensional solid hexahedral (brick) element with 8 nodes, reduced integration

and hourglass control. Each node has three degrees of freedom, translation in the x, y and z directions.

Reduced integration uses a lower-order integration to form the element stiffness. Reduced integration

reduces running time, especially in three dimensions. For first-order elements, the accuracy achieved

with full versus reduced integration is largely dependent on the nature of the problem (ABAQUS,

2007). The gage length model does not have any curved surfaces in which the stress state is of

interest. Therefore, a first-order element was chosen to mesh the part since it is computationally less

costly than a higher-order element. Analyses with both full and reduced integration were completed.

Since run time was on the order of a couple of minutes on a PC with a 3.2 GHz Pentium 4 processor,

93

not much difference in run times was observed when using elements with full or reduced integration.

Results were also not affected by the type of integration used. Therefore, it was decided to use

elements with reduced integration.

Hourglassing can be a problem with first-order, reduced integration elements in

stress/displacement analyses. Since the elements have only one integration point it is possible for

them to distort in such a way that the strains calculated at the integration points are all zero, which in

turn leads to uncontrolled distortion of the mesh, called hourglassing (ABAQUS, 2007). Therefore,

ABAQUS implemented hourglass control algorithms into the formulation of these elements.

Figure 6-3: ABAQUS meshed specimen gauge length.

In ANSYS, 448 SOLID185 elements were used to mesh the gauge length model, see Figure 6-

4. SOLID185 is a three-dimensional, first-order element with 8 nodes. Each node has three degrees

94

of freedom, translation in the x, y and z directions. Reduced integration with hourglass control was

also used with the SOLID 185 element.

Figure 6-4: ANSYS meshed specimen gauge length.

Mesh convergence studies are important because they ensure that the results obtained from a

numerical analyses were not dependent upon the model’s mesh. In an h-refinement mesh convergence

study, the number of elements is increased while certain output variables such as maximum von Mises

stress are monitored. In a p-refinement mesh convergence study, the order of the element is increased

while certain output variables are monitored.

An h-refinement mesh convergence study was conducted to ensure that results were not

dependent on mesh density. A p-refinement convergence study was not needed since higher-order

elements were not expected to improve the results for these analyses as previously discussed. Results

of the h-refinement mesh convergence study are shown in Figures 6-5 and 6-6.

95

343

344

345

346

347

348

349

350

0 200 400 600 800 1000

Number of Elements

Max

von

Mis

es S

tres

s [M

pa]

Figure 6-5: Mesh convergence using maximum von Mises stress.

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6

6.1

0 200 400 600 800 1000

Number of Elements

Max

Dis

plac

emen

t [m

m]

Figure 6-6: Mesh convergence using maximum displacement.

From the h-refinement mesh convergence study it was determined that 448 elements could

accurately predict the structural behaviour while maintaining computational efficiency. This was

96

determined because when increasing from 448 elements to 850, the maximum von Mises stress only

decreased by 1.3% and the maximum displacement only decreased by 2.4%. Since almost doubling

the number of elements had such a small effect on the change of predicted displacement and stress, the

solution is considered not to be dependent on the mesh.

6.1.2 Kachanov One-State Variable

The experimental creep rupture data presented in Section 4.6 showed that, under the

conditions tested, SA 455 exhibits tertiary creep behaviour. As summarized in the literature review in

Section 2.2, the Kachanov One-State Variable Technique is capable of modeling tertiary creep. The

uniaxial form of the One-State Variable equations, as introduced in Section 3.1.3, is:

mn

c tA ⎟⎠⎞

⎜⎝⎛−

=ω

σε1

'& (Equation 6-1)

( )mtB φ

χ

ωσω−

=1

'& (Equation 6-2)

where, cε& is the creep strain rate, A’, B’, n, m and Ø are material constants, σ is stress, ω is

creep damage and t is time.

These equations were implemented into ANSYS and ABAQUS via the creep user subroutine.

6.1.3 MPC Omega

As summarized in the literature review in Section 2.2, the MPC Omega Method is capable of

modeling tertiary creep. The uniaxial form of the MPC Omega Method, as introduced in Section

3.1.3, is:

97

cpeo

c εεε Ω= && (Equation 6-3)

1+Ω

Ω==

psc

psc

r

s

tt


εε&

& (Equation 6-4)

where, cε& is the creep strain rate, 0ε& is the initial creep strain rate (material constant), cε is

the creep strain, pΩ is the material constant omega, ts is time in service, tr is rupture time and cε& is the

effective creep strain rate.

These equations were implemented into ANSYS and ABAQUS via the creep user subroutine.

6.2 Results and Discussion

In order to validate the finite element analysis creep models, uniaxial creep runs were

conducted under the same loads and temperatures as the creep rupture experiments. The purpose of

this was to ensure that all derived creep material constants were correct and to verify if the MPC

Omega and Kachanov methods could accurately predict the measured creep behaviour of SA 455 steel

under uniaxial stress.

The results of the finite element analyses were compared to the experimental findings in

Figures 6-7 through 6-12. By inspecting these plots it is apparent that each of the four types of

analyses produced excellent failure time predictions. However, in terms of predicting true creep strain

at rupture, the MPC Omega method generally outperformed the Kachanov technique. It was found

that both ABAQUS and ANSYS commercial computer codes could model the high-temperature creep

ruptures.

98

The numerical analysis creep strain plotted in Figures 6-7 through 6-12 is the average creep

strain measured over the gage length calculated by dividing the total deformation along the gage

length by the final length of the gage length. This is how creep strain was measured in the lab and

thus to make an accurate comparison between numerical analysis results and experimental results,

creep strain must be measured in the same way.

Figures 6-13 and 6-14 were plotted to enable quick examination of the overall accuracy with

which the models predicted failure times and failure strains. Figures 6-15 and 6-16 show the typical

FEA creep damage contour plots that were produced by the analyses.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 500 1000 1500

Time (s)

True

Cre

ep S

trai

n

Experimental 266 MPaExperimental 296 MPaExperimental 327 MPaAnsys Kach 266 MPaAnsys Kach 296 MPaAnsys Kach 327 MPaAnsys Omega 266 MPaAnsys Omega 296 MPaAnsys Omega 327 MPaAbaqus Kach 266 MPaAbaqus Kach 296 MPaAbaqus Kach 327 MPaAbaqus Omega 266 MPaAbaqus Omega 296 MPaAbaqus Omega 327 MPa

Figure 6-7: SA 455 steel uniaxial creep validation of finite element analyses, T=550oC.

99

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1000 2000 3000 4000 5000

Time (s)

True

Cre

ep S

trai

n

Experimental 151 MPaExperimental 177 MPaExperimental 196 MPaAnsys Kach 151 MPaAnsys Kach 177 MPaAnsys Kach 196 MPaAnsys Omega 151 MPaAnsys Omega 177 MPaAnsys Omega 196 MPaAbaqus Kach 151 MPaAbaqus Kach 177 MPaAbaqus Kach 196 MPaAbaqus Omega 151Abaqus Omega 177Abaqus Omega 196


100

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 500 1000 1500 2000 2500 3000

Time (s)

True

Cre

ep S

trai

n


Figure 6-9: SA 455 steel uniaxial creep validation of finite element analyses, T=630oC

101

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 500 1000 1500 2000

Time (s)

True

Cre

ep S

trai

n



102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 500 1000 1500 2000 2500 3000

Time (s)

True

Cre

ep S

trai

n



103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1000 2000 3000 4000 5000 6000

Time (s)

True

Cre

ep S

trai

n



0

1000

2000

3000

4000

5000

6000

0 2000 4000 6000

Experimental Failure Time (s)

FEA

Pre

dict

ed F

ailu

re T

ime

(s)

Abaqus KachAbaqus MPCAnsys KachAnsys MPC

Figure 6-13: Predicted failure time.

104

0

10

20

30

40

50

60

70

80

0 20 40 60 80

Experimental Failure Strain (%)

FEA

Pre

dict

ed F

ailu

re S

train

(%)

Abaqus KachAbaqus MPCAnsys KachAnsys MPC

Figure 6-14: Predicted creep strain at failure.

Figure 6-15: Typical ABAQUS creep damage contour plot at failure.

105

Figure 6-16: Typical ANSYS creep damage contour plot at failure.

It should be noted that the numerical analyses predicted two necks to occur in the samples due

to the fact that the numerical analyses assumed the temperature over the gage to be constant and thus

necks formed at the location of peak stresses. As previously mentioned, the comparison of creep

strains presented in Figures 6-7 through 6-12 compared the averaged creep strain over the entire gage

length. During the experiments the sample necked at only one location, the location of peak

temperature. However, if gage length of the numerical analysis specimens was cut in half to account

for model symmetry then the strain measured over the gage length would be the same and only capture

one neck. Also, another solution to this discrepancy would be to analyze the gage lengths again with a

slightly higher temperature occurring at some location along the gage length, perhaps the middle, in

order to force only one neck to develop.

In order to further understand some of the formulation differences that exist between the

Kachanov technique and the MPC Omega method, plots of damage accumulation versus time are

shown in Figures 6-17 and 6-18. Examination of the plots revealed that when modeling high-

temperature creep, the Kachanov type equations showed an exponential type creep damage increase

while the MPC Omega method showed a linear creep damage increase. The Kachanov type equations

are more accurate at predicting the realistic trend in creep damage accumulation, while the MPC

Omega method damage parameter is more of a measure of life fraction consumed. Therefore, in most

106

cases, the MPC Omega method would be more numerically stable then the Kachanov technique when

modeling stress rupture.

Numerical instability is often encountered when modeling creep rupture due to the increasing

creep strain rates that lead to failure. The Kachanov creep strain rate is a function of creep damage,

see Equations 6-1 and 6-2, since the creep damage increases rapidly near failure, as plotted in Figure

6-17, numerical instability is an issue as creep damage increases past 50%. From Figures 6-7 to 6-12,

it is obvious that both the MPC Omega method and Kachanov technique can accurately predict the

creep rupture behaviour of SA 455 steel. However, the MPC Omega creep strain rate is not a function

of creep damage and so for high-temperature creep analysis, the MPC Omega method was found to be

more numerically stable.

Figure 6-17: Creep damage accumulation using the Kachanov technique; T=600oC S=177.3

MPa, solved with ABAQUS.

Figure 6-18: Creep damage accumulation using the MPC Omega method; T=600oC S=177.3

MPa, solved with ABAQUS.

107

Chapter 7 Numerical Pressure Vessel Analysis

Numerical pressure vessel analyses were conducted to gain a better understanding of the

vessel’s structural failure that occurred due to accidental fires. The effects of pressure vessel

geometry, fire intensity and type of fire impingement were studied. This study is of engineering

interest since it was shown experimentally that pressure vessels exposed to accidental fires can fail in

less than 30 minutes (Birk et al., 2006) and some design standards (CFR 49 Part 179, 2008; CGSB-

43.147, 2005; ASME,2001) require that pressure vessels exposed to accidental fires must resist failure

for at least 100 minutes. Both experimental and computational work will aid to improve the

understanding of pressure vessel failure and will hopefully lead to safer pressure vessel designs.

The numerical analysis work conducted to validate uniaxial creep rupture was performed with

both ANSYS and ABAQUS commercial FEA computer codes. However, when the problem was

complicated by the multiaxial states of stress present in the pressure vessel analyses, ABAQUS clearly

outperformed ANSYS. ANSYS was struggling with convergence issues when attempting to solve this

non-linear problem. Therefore, only ABAQUS was used to conduct all of the pressure vessel

analyses.

Implicit time integration was used for the creep analysis since it is unconditionally stable and

can tolerate relatively large time steps (Becker et al., 1993). For an implicit integration creep analysis,

the maximum allowable creep strain increment needs to be specified by the user. The smaller this

value, the more accurate the solution. However, a smaller maximum creep strain increment will

increase computational time as more increments will be needed. Therefore, the sensitivity of the

solution with respect to the maximum creep strain increment should be investigated. In ABAQUS, the

maximum creep strain increment is referred to as “CETOL”. From a sensitivity study, it was found

that when CETOL was varied between the recommended values of 1x10-3 and 1x10-4, the failure time

108

and run time were essentially unchanged. Since the value of CETOL did not affect the results it

indicated that the automatic time stepping was not being affected by the changing CETOL values.

Therefore, in order to keep the analysis as accurate as possible, the smallest value of CETOL, 1x10-4,

was used for all of the analyses.

Non-linear FEA solution techniques were used to perform the numerical pressure vessel

analyses. Sources of analysis non-linearities included the following:

• Geometric non-linearity due to large deformations

• Material non-linearity due to plastic and creep deformations.

Large deformations were present because this was a high temperature failure analysis of steel.

Structural vessel deformations were large enough to have a significant effect on the load-deflection

characteristics of the analysis. In general, to determine if large deformation behaviour should be

accounted for, practitioners use Roark and Young’s (1975) rule of thumb, which indicates that large

deformations should be accounted for if the transverse displacement exceeds about one-half the wall

thickness. Material plasticity was present in the analysis when temperatures and stresses were high

enough at one or more integration point(s) to cause material yielding. Plasticity was modeled as

bilinear isotropic because loading was monotonic and thus cyclic loading effects were not present.

Material creep was present in the analysis when temperatures at one or more integration point(s) were

above a certain pre-determined temperature (550oC in the case of this work). For further information,

Hinton (1992) provides a detailed discussion of the non-linearities associated with large deformation,

plasticity and creep.

All ABAQUS numerical analyses were submitted using script files written in Python and user

subroutines written in FORTRAN.

109

7.1 Geometry and Mesh

Numerical analyses were performed mainly on 500 US gallon pressure vessels. However,

some analyses were also conducted on 1,000 and 33,000 US gallon pressure vessels.

ABAQUS’ S4 finite shell elements were used to mesh the pressure vessels. The S4 shell

finite element has 4 nodes and six degrees of freedom at each node, three displacement components

and three rotation components. The element allows transverse shear deformation. It uses thick shell

theory as the shell thickness increases and becomes discrete Kirchhoff thin shell elements as the

thickness decreases.

The theory of thin shell structures was first developed by Kirchhoff (1850), Aron (1874) and

Love (1888). This theory was later refined by many researchers. The main condition that must be met

in order to use thin shell theory is that the ratio of thickness to radius of curvature of the structure must

be less than 0.1. The assumption of thin shell theory basically ignores the through thickness stresses

as they are usually at least about 5 times lower than the normal stresses in the other two directions.

This is similar to the assumption of plane stress, where one dimension of the structure is much smaller

than the other two, only it is applied to shell structures. The 500 and 1000 US gallon pressure vessels

both have thickness to radius of curvature ratios of 0.0148 and the 33,000 US gallon pressure vessel

has a ratio of 0.0105. All three pressure vessels thickness ratios are about of an order of magnitude

less than required to assume thin shell theory. Therefore, thin shell finite elements were used to mesh

the pressure vessels.

For shell finite element S4 in ABAQUS, the number of integration points through thickness

needs to be specified by the user. When the number of integration points through thickness was

changed from 3 to 5, it made no difference to the final solution. ABAQUS documentation suggests

that, for Gauss integration, 3 integration points be used through the thickness for conventional shell

110

elements. Therefore, the shell sections were defined to have 3 integration points through the

thickness.

Checking mesh sensitivity is an important step in ensuring that an analysis is efficient and

contains no significant mesh related errors. A p-refinement mesh convergence study was not possible

because ABAQUS does not have a higher-order thin shell element with large strain capabilities.

Therefore, an h-refinement mesh convergence study was conducted. The mesh in ABAQUS was

generated using free meshing. Therefore, the sole parameter that will affect the mesh size will be the

element size parameter. Decreasing the element size greatly increases run time since more elements

will be needed to mesh the vessel. When the element size was increased 50% from 20mm to 30mm,

for the 500 gallon pressure vessel, the predicted vessel failure time was only changed by 0.6%.

Therefore, to keep run times low, an element size of 30mm was used. Note that a 40mm element size

was attempted but the analysis became unstable since temperature gradients in elements became too

large because not enough nodes were present to accurately map the thermal gradients.

When specifying a through-thickness temperature gradient in ABAQUS for shell elements, the

user needs to decide on the number of temperature points through the thickness that will be used.

Temperatures were defined through the shell thickness at all node locations in a piecewise fashion.

This allowed for a through thickness temperature gradient to be established and the resulting loads

from this gradient were incorporated into the analyses. Increasing the number of temperatures

prescribed through the thickness increased the solution run time since more temperatures needed to be

stored in memory and more calculations needed to be made. When the number of temperatures

through the thickness was increased from 3 to 5, the solution did not change at all, however the

solution run time increased by 22%. Therefore, 3 temperatures through the thickness were used.

The engineering drawings of the 500 and 1000 US gallon pressure vessels, Figures B-1 and B-

2, indicate that the cylindrical part of the pressure vessel is made of SA 455 steel while the

111

hemispherical heads are made of SA 285-C. For this work it was assumed that the entire pressure

vessel was made of SA 455 steel. Since failure was never predicted numerically or experimentally in

the pressure vessel heads and since the materials have similar compositions, this assumption should

not affect the results. The tensile strength of SA 285-C is 380-515 MPa and its minimum yield

strength is 205 MPa (ASTM, 2007).

The engineering drawing of the 33,000 US gallon pressure vessel, Figure B-3, indicates that

the pressure vessel is made of TC-128 steel. This work only measured SA 455 steel’s creep rupture

data. Therefore, the 33,000 US gallon pressure vessel was modeled using SA 455 steel’s material

properties.

The following subsections introduce the various pressure vessel dimensions and meshes.

7.1.1 500 US gallon Pressure Vessel A detailed engineering drawing of the 500 US gallon pressure vessel is provided in Appendix

B as Figure B-1. The pressure vessel was modeled as a shell structure with 5 faces, see Figure 7-1.

Loads and boundary conditions are shown in Figures 7-2 and 7-3. The boundary conditions were set

to mimic a pressure vessel resting on two stands; the pressure vessel was only fixed in the axial

direction at one stand so as to prevent rigid body motion and to prevent the introduction of false axial

stresses. The details of the shell mesh are shown in Figures 7-4 and 7-5.

Figure 7- 1: 500 US gallon pressure vessel model.

112

Figure 7-2: 500 US gallon pressure vessel model with pressure loads shown as arrows.

Figure 7-3: 500 US gallon pressure vessel model, view of underside; boundary conditions on the left fixed all translational degrees of freedom and rotation about the 2 and 3 directions,

boundary conditions on the right fixed translation in the 1 and 2 directions.

113

Figure 7-4: 500 US gallon pressure vessel mesh using 12,848 ABAQUS S4 finite shell elements.

Figure 7-5: 500 US gallon pressure vessel mesh, zoom in view of end detail.

114

7.1.2 1,000 US gallon Pressure Vessel

A detailed engineering drawing of the 1000 US gallon pressure vessel is provided in Appendix

B as Figure B-2. It has the same dimensions as the 500 US gallon pressure vessel except that it is

about twice as long. The pressure vessel was modeled as a shell structure with 3 faces, see Figure 7-6.

Loads and boundary conditions are shown in Figure 7-7. Pressure loads on faces are represented by

the pink arrows. The boundary conditions were set to mimic a pressure vessel resting on two stands;

the pressure vessel was only fixed in the axial direction at one stand to prevent rigid body motion and

to prevent the introduction of false axial stresses. The shell element mesh is shown in Figure 7-8. The

detail of the mesh on the hemispherical head is similar to that shown in Figure 7-5.


115

Figure 7-7: 1000 US gallon pressure vessel model, view of underside; boundary conditions on the left fixed all translational degrees of freedom and rotation about the 2 and 3 directions,

boundary conditions on the right fixed translation in the 1 and 2 directions.

Figure 7-8: 1000 US gallon pressure vessel mesh using 20,048 ABAQUS finite shell elements.

116

7.1.3 33,000 US gallon Pressure Vessel

A detailed engineering drawing of the 33,000 US gallon pressure vessel is provided in

Appendix B as Figure B-3. The 33,000 US gallon pressure vessel is manufactured with elliptical

heads. However, for the sake of simplicity, in this analysis the heads were modeled as being

hemispherical. This geometry change should not affect the results much since failure occurs far from

the heads of the pressure vessels. The pressure vessel was modeled as a shell structure with 3 faces,

see Figure 7-9. Loads and boundary conditions are shown in Figure 7-10. Pressure loads on faces are

represented by the pink arrows. The boundary conditions were set to mimic a pressure vessel resting

on two stands; the pressure vessel was only fixed in the axial direction at one stand to prevent rigid

body motion and to prevent the introduction of false axial stresses. The shell element mesh is shown

in Figure 7-11. The detail of the mesh on the hemispherical head is similar to that shown in Figure 7-

5.


117

Figure 7-10: 33000 US gallon pressure vessel model, view of underside; boundary conditions on

the left fixed all translational degrees of freedom and rotation about the 2 and 3 directions, boundary conditions on the right fixed translation in the 1 and 2 directions.

Figure 7-11: 1000 US gallon pressure vessel mesh using 24,946 ABAQUS S4 finite shell elements.

118

7.2 Creep Damage Models

7.2.1 Kachanov One-State Variable Analyses

The Kachanov One-State Variable Technique was shown to accurately predict the uniaxial

creep behvaiour of SA 455 steel in Section 6. In order to use the technique to model the creep

deformation of pressure vessels, the equations were modified, as discussed in Section 3.1.2, to capture

the effects of multiaxial states of stress.

The multiaxial forms of the one-state variable equations are:

mijn

cij t

SA

σωσε ⎟

⎠⎞

⎜⎝⎛−

=1

'23

& (Equation 7-1)

( )mr tB φ

χ

ωσ

ω−

=1

'& (Equation 7-2)

( )σαασσ −+= 11r (Equation 7-3)

where, cijε& is the creep strain rate component, A’, B’, n, m and Ø are material constants, σ is

the effective stress, ω is creep damage, Sij is the deviatoric stress component, t is time, rσ is rupture

stress, 1σ is the maximum principal stress and α is the multiaxial behaviour material constant.

In order to find the value of creep constant α, creep rupture experiments of notched tensile

specimens need to be conducted as discussed by Hyde et al. (2006). These data were unavailable since

no notched creep rupture specimens were tested. However, in the absence of these data, Equation 7-4

was used, as suggested by Lemaitre and Chaboche (1985).

( ) ( )2

12

213132

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞⎜

⎝⎛−++= σσννσσ m

r (Equation 7-4)

where, ν is poison’s ratio and mσ is the hydrostatic stress.

119

These equations were implemented into ABAQUS via a creep user subroutine.

7.2.2 MPC Omega Analyses

The MPC Omega Method was shown to accurately predict the uniaxial creep behvaiour of SA

455 steel in Section 6. In order to use the technique to model the creep deformation of pressure

vessels, the equations were modified, as discussed in Section 3.1.2, to capture the effects of multiaxial

states of stress.

The multiaxial forms of the MPC Omega equations are:

ctubee

Sijcij

εεσ

ε Ω⋅⋅= 023

&& (Equation 7-5)

1+ΩΩ

==tubes

ctubes

c

r

s

tt


εε&

& (Equation 7-6)

where, cijε& is the creep strain rate component, 0ε& is the initial creep strain rate (material

constant), cε is the creep strain, tubeΩ is the material constant omega (see Equation 3-40), Sij is the

deviatoric stress component, σ is effective stress, ts is time in service, tr is rupture time and cε& is the

effective creep strain rate.

These equations were implemented into ABAQUS via a creep user subroutine.

7.3 Loading Conditions

From the literature review conducted in Section 2.3 it was shown that the thermal response of

a pressure vessel exposed to an accidental fully engulfing or partially engulfing fire is well understood.

120

The main objective of this work was to develop a FEA model capable of numerically predicting the

high temperature structural failure of a pressure vessel. Therefore, it was beyond the scope of this

work to model the detailed heat transfer that occurs between the vessel contents, the vessel and the

fire. Rather, the pressure vessel temperature distribution data experimentally gathered in the literature

review was used to apply thermal loads at nodes as temperatures during a static structural analysis.

The relative reliability of the numerical model can be examined with experimental results and

various loading conditions of interest can be investigated. First, to assess the general performance of

the numerical analysis model, a time independent temperature and pressure distribution loading will be

examined. Then, to investigate the affect of local hot spots on pressure vessel failure, time

independent hot spots will be applied.

7.4 Time Independent Fully Engulfing Fire Loading

7.4.1 Description of Fully Engulfing Fire Loading

The loading discussed in this section was applied to 500, 1000 and 33000 US gallon pressure

vessels.

There exists substantial variability in the exact temperature distribution of a pressure vessel

exposed to a fully engulfing fire. Many factors such as wind, fire intensity and surroundings of the

pressure vessel cause variations in temperature distribution. Variability in the internal pressure history

exists since pressure is dependent on the heating distribution on the vessel and also it is controlled by a

pressure relief valve (PRV). PRVs have been shown to have highly variable performance by

Pierorazio and Birk (1998). Therefore, it was decided not to exactly model the complex loading

history that a pressure vessel is exposed to during an accidental fire. Rather, the vessel loading was

121

simplified to consist of a time independent temperature distribution and internal pressure. This was

achieved by ramping up the vessel temperature distribution and internal pressure at the beginning of

the numerical analysis during a time independent load step. After that, temperatures, internal pressure

and fill level were held constant during the remainder of the simulation.

Depending on the fill level of a pressure vessel, a certain percentage of the vessel volume will

be occupied with liquid and the remainder will be occupied with vapour. The liquid, being more

dense, will occupy the bottom portion of the vessel. In reality the fill level changes with time. Before

the PRV opens the liquid level will rise due to liquid thermal expansion. The fill level will drop once

the PRV is activated since vapour will leave the vessel as the lower internal pressure causes the liquid

to boil. This complexity was removed from the present numerical analysis and the fill level was

considered to stay constant at 50%. Constant fill levels of 30% and 70% were also analyzed and

produced similar failure times to the 50% fill. This is because moving the large thermal gradient up

and down between 70% and 30% fill does not change the stress state in the failure zone much.

Data from fire tests summarized in the literature review, Section 2.3, show that the liquid

wetted wall stays below about 130oC during a fully engulfing fire, while temperatures of the vapour

wetted wall will rise to anywhere between 550oC to 750oC depending on the intensity of the fire as

well as other external factors. The liquid wetted wall remains much cooler because the liquid propane

is much more efficient at removing heat from the vessel wall than the vapour is. This is because liquid

propane has a much higher specific heat, thermal conductivity and density and also the boiling of

liquid propane will absorb much heat.

When the PRV opens and vapour is vented, there will be boiling in the liquid and this will

result in a frothing layer at the liquid-vapour interface consisting of a 2-phase liquid-vapour mixture as

the liquid surface bubbles. This frothing layer acts to spread out the severe temperature gradient at

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that location in the vessel wall. The frothing layer is also difficult to define precisely and has been

simplified for this numerical analysis.

Figure 7-12 shows how various angles are defined and measured for this analysis in order to

impose the discussed temperature distribution on the pressure vessels.

Figure 7-12: Cross section representation of pressure vessel temperature distribution showing angle measurement convention.

In this study, the size of the frothing region was defined by a constant angle β, where β is the

angle that spans the frothing region. A sensitivity study, see Figure 7-13, revealed that when β was

increased from 10º to 30o, the predicted vessel failure time only increased by 4%. Therefore, using

engineering judgment and from a desire to keep the analysis stable, β was set equal to 20º.

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705

710

715

720

725

730

735

740

745

0 5 10 15 20 25 30 35

β (deg)

Fail

Tim

e (s

)

Figure 7-13: Sensitivity study of size of frothing region.

When pressure vessels are exposed to engulfing fires, they consistently fail at the location of

peak wall temperature, somewhere in the vapour space wall region. In this study it was assumed that

the peak wall temperature occurred on the top of the vessel. In order to achieve a peak wall

temperature on the top of the vessel, temperature gradient, δ, is defined. The physical interpretation of

δ is that for every degree traveled from the top of the frothing region towards the top of the vessel, the

temperature increases by the value of δ. A sensitivity study, see Figure 7-14, on the value of δ

revealed that increasing δ by a factor of 4 caused a 6% increase in vessel failure time. A δ value of 0.1

oC/deg was found to be most appropriate since it predicted failure at the top of the vessel, yet was low

enough to keep the temperature almost constant throughout the vapour wetted region of the vessel,

behaviour typically observed in experimental pressure vessel fire tests.

124

700

710

720

730

740

750

760

0 0.05 0.1 0.15 0.2 0.25

δ [oC/deg]

Fail

Tim

e (s

)

Figure 7-14: Sensitivity study of value of thermal gradient in the vapour wall space.

In the finite element analysis model, temperatures were specified as temperature loads at

nodes via a piecewise function using the UTEMP user subroutine. The temperature in the vapour

wetted region was described by a linear function, see Equation 7-2. The temperature in the frothing

region was described by a cosine function that smoothes out the large temperature gradient, see

Equation 7-3. The temperature in the liquid wetted region was constant, see Equation 7-4. The cross

sectional temperature distribution is plotted in Figure 7-15 and the 3D temperature distribution of the

vessel is shown in Figure 7-16.

δθ ⋅−= THTEMP (Equation 7-2)

125

( )( )[ ] ( )( )

( )( ) TCfillTCTH

fillTCTHfillTEMP

+⎥⎦⎤

⎢⎣⎡ −−−

+

⎥⎦⎤

⎢⎣⎡ −−−⋅+−=

2*

2*/180cos

βδ

βδββθ o

(Equation 7-3)

TEMP = TC (Equation 7-4)

where, θ is the current angle

fill is the angle to the bottom of the frothing region (measured from top centre)

β is the angle of the frothing region (measured from bottom of frothing region)

δ is the gradient of temperature rise in the vapour wetted wall.

TH is the maximum temperature in the vapour wetted wall.

TC is the temperature of the liquid wetted wall (130oC).

Also, during fire exposure, it is known that a through-thickness temperature gradient exists.

This was analytically calculated to be about 14oC for the liquid wetted wall and 6oC for the vapour

wetted wall and experimentally by Moodie et al. (1988). This through-thickness temperature gradient

was incorporated into the finite element analyses.

0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180

Theta (deg)

Tem

pera

ture

(o C) Frothing

RegionVapour Wetted

Liquid Wetted

Figure 7-15: Piecewise temperature distribution; theta is measured from top centre of

vessel. δ=0.1, β=20o, TH=650oC, TC=130oC, fill =50% or 90o.

126

Figure 7-16: Imposed vessel temperature distribution (oC) of 500 US gallon pressure vessel, with δ=0.1, β=20o, TH=650oC, TC=130oC, fill =50% or 90o.

7.4.2 Results and Discussion of the Fully Engulfing Fire Loadings

7.4.2.1 500 US gallon Pressure Vessel

The analyses were conducted with the temperature distribution discussed in Section 7.4.1,

and parameter settings of δ=0.1, β=20o, TC=130oC and fill = 50% or 90o. The numerical analysis

results of the 500 US gallon pressure vessels are summarized in Table 7-1 and Figure 7-17. It has

been shown experimentally by Birk et al. (1994) that a pressure vessel in a fully engulfing fire

will empty in about 15 minutes via the pressure relief valve. Therefore, the numerical analysis

was focused on loading conditions that predict pressure vessel failure times of less than about 20

minutes. This is why an analysis was not conducted for a maximum temperature of 620oC and an

internal pressure of 1.90MPa (275 psig). Pressure vessel failure, in all cases except one, was

127

determined to occur due to creep damage exhaustion; that is, creep damage values surpassed 0.99

at many locations along the top of the pressure vessels.

Table 7-1: MPC Omega numerically predicted failure times [min] of pressure vessels with respect to internal pressure and maximum temperature solved with ABAQUS.

P [MPa] (psig)

T [oC] 1.90MPa (275 psig)

2.07MPa (300 psig)

2.24MPa (325 psig)

680 4.4 2.6 1.1*

650 21.9 12.1 7.1

620 X 71.9 39.1

*Failure due to combined plastic and creep deformation.

0

10

20

30

40

50

60

70

80

600 620 640 660 680 700

Maximum Temperature [oC]

Failu

re T

ime

[min

]

p=1.90 MPa(275psig)p=2.07 MPa (300psig)p=2.24 MPa (325psig)

Figure 7-17: 500 US gallon pressure vessel predicted failure times (MPC Omega,

ABAQUS).

For the loading case of a maximum temperature of 680oC and an internal pressure of

2.24MPa (325 psig), there was plastic deformation in the failure zone. The combined plastic and

creep failure was structurally unstable and thus computationally difficult to solve. This meant

128

that the predicted failure time for this loading condition was determined by ABAQUS’ inability

to continue the analysis no matter how small the time step size.

All predicted failures, except T=680oC p=2.24MPa (325 psig), were similar to that shown

in Figures 7-18, 7-19 and 7-20, occurring along an axial line on the top of the pressure vessel with

no plastic deformation occurring in the failure zone. Therefore, these failures could be

categorized as being mostly driven by stress rupture high-temperature creep. Plastic deformation

only occurred in the vicinity of the frothing region. This was because the large temperature

gradient through the frothing region produced large thermal stresses and thus plastic strains.

In the legend of the FEA contour plots, “fraction=0.775” indicates that the values at the

integration points closest to the outer surface are being plotted. “SDV1” is creep damage,

“CEEQ” is effective creep strain, “PEEQ” is effective plastic strain, “S” is stress and “NT13” is

the nodal temperature on the outer surface of the pressure vessels.

Predicted MPC Omega creep damage and effective creep strain contour plots for all cases

are shown in Appendix C.

129

Figure 7-18: Predicted MPC Omega creep damage at time of failure, T=650oC p=2.07MPa

(300 psig); solved with ABAQUS.

Figure 7-19: Predicted MPC Omega effective creep strain at time of failure, T=650oC

p=2.07MPa (300 psig); solved with ABAQUS.

130

Figure 7-20: Predicted MPC Omega effective plastic strain at time of failure, T=650oC


The relatively fast failure times indicated that tertiary creep must be the dominant

phenomena. Examination of Figure 7-21 confirmed that creep was indeed tertiary. Stress

redistribution or relaxation is a much studied and widely known phenomenon that occurs with

creep deformation (Hinton, 1992). Figure 7-22 shows how fast stress redistribution occurred with

tertiary creep and how unstable it became with the onset of failure. In a fully engulfing fire, the

stress in the failure zone quickly redistributed itself early on and then remained almost constant

for the remainder of the vessel life until it dropped quickly with failure. Since the fire is fully

engulfing, the wall temperature everywhere in the vapour space is near the peak wall temperature.

This meant that a significant portion of the pressure vessel was softened and thus not much stress

redistribution could occur during the course of the analysis since load equilibrium must be

maintained. This is shown in Figure 7-22, as the stress level remained almost constant for most

of the analysis. The extreme drop in stress near the end of the analysis showed that material on

131

the verge of failure will experience a vast reduction in its load bearing capacity. This

phenomenon has been thoroughly studied and presented by other researchers as well (Becker,

2002).

Figure 7-21: Time history of MPC Omega effective creep strain at point of eventual failure,

T=650oC p=2.07MPa (300 psig); solved with ABAQUS.

Figure 7-22: Time history of effective stress at point of eventual failure, T=650oC


The numerically predicted failure times are within expectations based on fire test results

of Birk et al. (2006). Birk conducted partially engulfing fire tests of thermally protected 500 US

gallon propane pressure vessels that were identical to those modeled in this study. In those tests

132

the tank wall temperature, pressure and fill level were changing with time and therefore direct

comparisons cannot be made. However, they observed failure times in the range of 10 to 15

minutes of vessels averaging peak wall temperatures and internal pressures of about 650oC and

2.07MPa (300 psig). Also, from experimental fire test data, it is expected that for a maximum

wall temperature of 650oC, the failure edge would be relatively thick and failure would be mostly

due to high-temperature creep. Figure 7-23 shows the shape of a vessel failure that was fire

tested with a 25% engulfing fire. The shape of the failure is very similar to that shown in Figure

7-19; although there is not as much vertical displacement predicted because the force of the

exhausted fuel jet was not modeled. Therefore, it is believed that the results of this study are

realistic since both failure time and failure mode were well predicted.

Figure 7-23: Failed fire tested 500 US gallon pressure vessel; 25% engulfing fire, peak wall temperature = 740oC, peak internal pressure = 2.59MPa (375 psig).

133

The same loading conditions summarized in Table 7-1 were also simulated using the

Kachanov One-State Variable Technique. The One-State Variable Technique proved to be more

unstable than the MPC Omega method for this application. This instability was expected as

discussed in Section 6.2 due to the quick increase in creep damage that occurred near failure.

These analyses consistently ended with convergence errors before creep damage values were at a

critical level. For this reason not all loading cases were solved using the One-State Variable

Technique. Table 7-2 summarizes the results found using the One-State Variable Technique.

Failure for the One-State Variable Technique is predicted when the analyses became numerically

unstable.

Table 7-2: Kachanov One-State Variable Technique numerically predicted failure times [min] of pressure vessels with respect to internal pressure and maximum temperature

solved with ABAQUS.

P [MPa] (psig)

T [oC] 2.07 MPa (300 psig)

2.24 MPa (325 psig)

650 10.8 6.2

620 X 30.4

The Contour plots of creep damage, effective creep strain and effective plastic strain at

time of failure, found using the One-State Variable Technique, are plotted in Figures 7-24, 7-25

and 7-26. From Figure 7-17, it is seen that for the T=650, p=2.07MPa (300 psig) loading, the

analysis becomes unstable when maximum creep damage is about 28%. Note that since the creep

damage for the One-State Variable Technique does not increase linearly with time this does not

mean that about 72% of the life time remains. From examining Figure 6-17, it is clear that for

uniaxial creep loading, a Kachanov One-State Variable creep damage value of about 28%

indicates that only about 10-15% of the component lifetime remains. Therefore, this shows that

134

the One-State Variable technique becomes unstable in these analyses when about 10-15% of

pressure vessel life remains.

Figure 7-24: Predicted One-State Variable Technique creep damage at time of failure, T=650oC p=2.07MPa (300 psig); solved with ABAQUS.

Figure 7-25: Predicted One-State Variable Technique effective creep strain at time of failure, T=650oC p=2.07MPa (300 psig); solved with ABAQUS.

135

Figure 7-26: Predicted One-State Variable Technique effective plastic strain at time of

failure, T=650oC p=2.07MPa (300 psig); solved with ABAQUS.

7.4.2.2 1,000 US gallon Pressure Vessel

The 1000 US gallon pressure vessel was analyzed to examine the effects of vessel

dimensions on failure time. The 1000 US gallon pressure vessel was identical to the 500 US

gallon pressure vessel except that it is about twice as long. The 500 US gallon pressure vessel

has a length to diameter ratio of 3.26 while the 1000 US gallon pressure vessel has a length to

diameter ratio of 5.53. The 1000 US gallon pressure vessel was solved exclusively with MPC

Omega creep damage method since it was shown to be more robust than the One-State Variable

Technique in Section 7.4.2.1. For a loading condition in which the maximum temperature was

650oC and the internal pressure was 2.07MPa (300 psig), the 1000 US gallon pressure vessel

failed in 10.2 minutes, this is 1.9 minutes sooner then the 500 US gallon pressure vessel failed

under the exact same loading conditions, see Table 7-3. The shorter 500 US gallon pressure

vessel takes longer to fail then the longer 1000 US gallon pressure vessel because of end effects.

136

Since the 1000 US gallon pressure vessel is longer it is easier to bend and thus a higher state of

stress is developed in the failure zone. This higher state of stress in the failure zone will cause

quicker failures. Therefore, it was shown that pressure vessels with a lower length to diameter

ratio are inherently safer when exposed to accidental fires.

The creep damage and creep strain contour plots are shown in Figures 7-27 and 7-28.

The 1000 US gallon pressure vessel’s failure zone did not contain any plastic strain, see Figure 7-

29. Therefore, the failure was driven by creep deformation and can be classified as a stress

rupture high-temperature creep failure.

Figure 7-27: 1000 US gallon vessel predicted MPC Omega creep damage at time of failure, T=650oC p=2.07MPa (300 psig); solved with ABAQUS.

137

Figure 7-28: 1000 US gallon vessel predicted MPC Omega effective creep strain at time of failure, T=650oC p=2.07MPa (300 psig); solved with ABAQUS.

Figure 7-29: 1000 US gallon vessel predicted MPC Omega effective plastic strain at time of failure, T=650oC p=2.07MPa (300 psig); solved with ABAQUS.

138

Table 7-3: Comparison between 500 and 1000 US gallon analysis results; peak wall temp = 650oC, p=2.07MPa (300 psig).

Pressure Vessel Size [US gallon]

Length / Diameter

Thickness / Diameter

Fail Time [min]

Length of Failure Zone [mm]

Length of Failure Zone / Diameter

500 3.26 0.0074 12.1 450 0.472

1000 5.53 0.0074 10.2 540 0.566

Table 7-3 shows that the length of the failure zone increased by 20% when the length of

the pressure vessel was increased by about a factor of 2. The length of the failure zone is

important since it is often used to examine whether or not a BLEVE occurred. Investigating if a

BLEVE failure or jet release failure occurred is beyond the scope of this work since the internal

fluid was not modeled. Predicting structural failure of the vessels was the current objective.

However, the failure length data could be useful to other researchers in the field.

7.4.2.3 33,000 US gallon Pressure Vessel

The 33000 US gallon pressure vessel was analyzed to examine the effects of vessel

dimensions on failure time. The 33000 US gallon pressure vessel has the same length to diameter

ratio as the 1000 US gallon pressure vessel; however, it is about 3.5 times larger with respect to

some of the major dimensions. Also, the 33000 US gallon pressure vessel has a smaller thickness

to diameter ratio than the 500 and 1000 US gallon pressure vessels. The 33000 US gallon

pressure vessel was solved exclusively with MPC Omega creep damage method since it was

shown to be more robust than the One-State Variable Technique in Section 7.4.2.1. For a loading

condition in which the maximum temperature was 620oC and the internal pressure was 2.07MPa

(300 psig), the 33000 US gallon pressure vessel failed in 4.85 minutes, this is 67.1 minutes faster

then the 500 US gallon pressure vessel failed under the exact same loading conditions. This huge

139

difference in failure time can be explained by two key geometrical differences, the larger length

to diameter ratio and the smaller thickness to diameter ratio. As shown by the results of the 1000

US gallon pressure vessel, a longer pressure vessel will fail quicker than a shorter one with the

same diameter. The thickness to diameter ratio of the 500 and 1000 US gallon pressure vessels

was 0.0074, compared to 0.0052 for the 33000 US gallon pressure vessel. This 30% reduction in

thickness to diameter ratio creates a 30% increase in vessel wall stress and thus has a great effect

on the fire survivability of a pressure vessel exposed to a fully engulfing fire. The creep damage

and creep strain contour plots are shown in Figures 7-30 and 7-31. The 33000 US gallon pressure

vessel’s failure zone contained plastic strain, see Figure 7-32. Therefore, the failure was driven

by both creep and plastic deformation and can be classified as a combination of high-temperature

creep and short-term overheating stress rupture.

Figure 7-30: 33000 US gallon vessel predicted MPC Omega creep damage at time of failure, T=620oC p=2.07MPa (300 psig); solved with ABAQUS.

140

Figure 7-31: 33000 US gallon vessel predicted MPC Omega effective creep strain at time of failure, T=620oC p=2.07MPa (300 psig); solved with ABAQUS.

Figure 7-32: 33000 US gallon vessel predicted MPC Omega effective plastic strain at time of failure, T=620oC p=2.07MPa (300 psig); solved with ABAQUS.

141

Table 7-4: Comparison between 500 and 33000 US gallon analysis results; peak wall temp = 620oC, p=2.07MPa (300 psig).

Pressure Vessel Size [US gallon]

Length / Diameter

Thickness / Diameter

Fail Time [min]

Length of Failure Zone [mm]

Length of Failure Zone / Diameter

500 3.26 0.0074 67.1 360 0.377

33000 5.53 0.0052 4.85 3500 1.148

Therefore, the length of the failure zone normalized by the diameter of the pressure

vessel increased by 205% when the length of the pressure vessel normalized by the diameter of

the vessel was increased by about a factor of 2 and the thickness of the vessel normalized by the

diameter of the vessel was decreased by 30%. The length of the failure zone is important since it

is often used to examine whether or not a BLEVE occurred. Thus it was concluded from Table 7-

3 and 7-4 that the length of the failure zone was more sensitive to a change in vessel thickness

then it was to a change in vessel length.

7.5 Time Independent Hot Spots

It is worth examining the structural behaviour of pressure vessels exposed to local hot

spots since not all accidental fire impingements are fully engulfing. For example, a nearby pipe

could rupture and cause a jet fire to come in contact with a pressure vessel which would lead to a

localized hot spot. Another example is that a thermally protected pressure vessel in a fully

engulfing fire could have a defect in its thermal protection, which would also lead to a localized

hot spot. The analyses were conducted as time independent because the heat up time will vary on

many external factors such as fire intensity and wind. Therefore, the analyses considered steady

state thermal conditions and predicted failure time as the time needed to achieve failure once

142

steady state conditions were reached. This is considered a valid assumption of loading since

minimal creep or plastic deformation will occur during heat up due to low wall temperatures.

The hot spot analyses were only conducted on 500 US gallon pressure vessels because

the bulk of the local fire impingement experimental fire testing done by Birk et al. was on 500 US

gallon pressure vessels.

7.5.1 Description of Hot Spot Loading

The hot spot was placed on the top dead centre of the pressure vessel. The shape of the

hot spot is a circle projected onto the top dead centre of the pressure vessel. This most closely

simulated the case of a jet fire impingement. The size of the hot spot was increased until failure

of the pressure vessel was observed within the time window of this work, about 20 minutes. It

was found that a hot spot of radius 100mm generated such failure times at wall temperatures

known to be achieved by a pressure vessel exposed to fire impingement, about 700oC.

The MPC Omega method was used to compute creep strain and creep damage for the hot

spot loadings. The MPC Omega method was shown to be more robust than the One-State

Variable technique when analyzing pressure vessel stress rupture in Section 7.4.2.

In order to apply the thermal loading as temperatures at nodes, the pressure vessel face

needed to be partitioned into various areas, both for applying temperature loads and for ease of

meshing, see Figures 7-33, 7-34 and 7-35. The partitioning of faces and meshing of the pressure

vessel varied based on the size of the hot spot. The partitioned faces and meshes shown in

Figures 7-33, 7-34 and 7-35 are for a 100mm radius hot spot. The partitioned faces and meshes

for the 50mm and 75mm hot spots are not shown because they are simply smaller versions of

what is shown in Figures 7-33, 7-34 and 7-35.

143

Figure 7-33: 500 US gallon pressure vessel part model for hot spot analysis (100mm radius).

Figure 7-34: 500 US gallon pressure vessel mesh for hot spot analysis (100mm radius) using 14,058 ABAQUS shell finite elements.

144

Figure 7-35: Detailed view of top dead centre of 500 US gallon pressure vessel mesh for hot spot analysis (100mm radius).

The temperature distributions were imposed based on experimentally measured pressure

vessel hot spots (Birk et al., 2003). The experimentally measured hot spots did not produce

pressure vessel failure. This was because the test was not meant to analyze pressure vessel failure

but rather to map the temperature distribution that a local hot spot induced in the pressure vessel

wall. Therefore, the measured temperature distribution, see Figure 7-36, was scaled up in order

to impose temperatures high enough to cause failure.

145

Position (cm)

Tem

pera

ture

(°C

)

0 10 20 30 40 500

25

50

75

100

125

150

175

200

Fire Test 4:Jacket, Insulation, 7.6 cm (3 in) Defect

20 min

15 min10 min

0 min

5 min

Figure 7-36: Experimentally measured temperature distribution of a pressure vessel with a

7.6 cm defect (Birk and VanderSteen, 2003).

7.5.2 Results and Discussion of Time Independent Hot Spot Loading

A summary of the hot spot analysis results is presented in Table 7-5. Results of the hot

spot analysis with a 100 mm radius and 720oC maximum wall temperature are shown in Figures

7-37 through 7-46. The results of the other hot spot analyses are presented in Appendix D.

Table 7-5: 500 US gallon pressure vessel hot spot predicted failure times [min].

Maximum wall temperature [oC]

Hot spot radius [mm] 680 700 720

50 * * *

75 * 58.2 24.1

100 55.6 20.4 9.7

*Failure not observed in a 100 minute analysis.

146

Note that the hot spot radius referred to in these analyses refers to the radius of the inner,

smallest circle, which contains the peak wall temperatures. However, Birk and VanderSteen

(2003) found it more convenient to deal with the size of the defect in the thermal insulation

during their experimental work. Therefore, to introduce the thermal defect size and bridge the gap

between the numerical and experimental work terminology, Table 7-6 was created.

Table 7-6: Relationship between hot spot radius and thermal defect radius.

Hot Spot Radius [mm] Thermal Defect Radius [mm]

50 125

75 187.5

100 250

It was apparent from Table 7-6 that the thermal defect radius was always 2.5 times

greater then the hot spot radius. This was the approximate trend observed experimentally by Birk

and VanderSteen (2003) and thus it was used to relate the values.

0

10

20

30

40

50

60

70

80

600 650 700 750

Maximum Temperature [oC]

Failu

re T

ime

[min

]

Fully Engulfed p=1.90MPa(275psig)Fully Engulfed p=2.07MPa(300psig)Fully Engulfed p=2.24MPa(325psig)Hot Spot r=75mm,p=2.07MPa (300psig)Hot Spot r=100mm,p=2.07MPa (300psig)

Figure 7-37: Comparison between fully engulfing fire and hot spot failure times.

147

Figure 7-38: 500 US gallon pressure vessel, imposed temperature distribution [oC] for hot spot analysis (100mm radius); maximum temperature of 720oC, pressure of 2.07MPa (300

psig), solved using ABAQUS.

Note that since this is a hot spot analysis, the temperature of the liquid wetted wall was

50oC. This is lower then the 130oC liquid wetted wall temperature used in the fully engulfing fire

analyses. The wall temperature was lower since the hot spot was applied on the vapour wetted

region of the vessel, so there was no direct contact between the fire and the liquid wetted wall.

The value of 50oC was obtained from experimental observations made by Birk throughout his

pressure vessel fire testing.

148

Figure 7-39: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of MPC Omega creep damage at time of failure; maximum temperature of 720oC, pressure of

2.07MPa (300 psig), solved using ABAQUS.

Figure 7-40: 500 US gallon pressure vessel hot spot analysis (100mm radius), time history of MPC Omega creep damage at location of eventual failure; red plot is inner surface, green

plot is outer surface; maximum temperature of 720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

149

All the calculated creep damage occurred near the hot spot. This was because creep

damage can only occur in regions where creep deformation occurs. For SA 455 steel, creep

deformation was determined as being significant at temperatures above 550oC. Therefore, the

only regions that could contain creep damage and creep deformation were regions in which the

temperature was above 550oC.

The creep damage plot in Figure 7-40 shows that for hot spot analyses the creep damage

rate in the failure region decelerated throughout the analysis. This deceleration of creep damage

was caused by the large and continuous stress redistribution that occurred during the hot spot

analysis. This stress redistribution acted to redistribute load away from the hot spot region to

stiffer, cooler, regions, Figure 7-47.


MPC Omega effective creep strain at time of failure; maximum temperature of 720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

150

Figure 7-42: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of MPC Omega effective creep strain at time of failure; zoomed in on failure zone; maximum

temperature of 720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

Figure 7-43: 500 US gallon pressure vessel hot spot analysis (100mm radius), time history of MPC Omega effective creep strain at location of eventual failure; red plot is inner surface,

green plot is outer surface; maximum temperature of 720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

151

From examination of Figure 7-43, it was clear, as expected, that creep in the failure zone

was predominantly characterized by tertiary creep.


effective plastic strain at time of failure; maximum temperature of 720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

As shown in Figure 7-44, pressure vessels exposed to local fire impingements

experienced plastic strains in the failure region. Therefore, failure was characterized by a

combination of creep and plastic deformation.

152


hoop stress at time of failure; maximum temperature of 720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

Figure 7-46: 500 US gallon pressure vessel hot spot analysis (100mm radius), contour plot of hoop stress at time of failure; zoomed in on failure region; maximum temperature of 720oC,

pressure of 2.07MPa (300 psig), solved using ABAQUS.

153

Due to creep deformation induced stress redistribution, the stress in the failure region was

continuously lowered as the failure region deformed.

Figure 7-47: 500 US Gallon pressure vessel hot spot analysis (100mm radius), time history

of von Mises stress at location of eventual failure; red plot is inner surface, green plot is outer surface; maximum temperature of 720oC, pressure of 2.07MPa (300 psig), solved

using ABAQUS.

Direct comparisons with experimental fire tests were not possible because the analyzed

local fire impingements were of different size and shape than those fire tested. Also, during the

fire tests, there was a heat up time as well as fluctuations in temperature and pressure due to

variations in wind and fire intensity which made a direct comparison not possible. However, the

failure times were in the range expected as experimentally tested pressure vessels with 8% and

16% thermal defects failed in the range of 22 – 40 minutes (Birk et al., 2006) with peak wall

temperatures comparable to values used in the analyses. The shape of an experimentally

measured failure region is shown in Figure 7-48. Note that the failure region was smaller than

those measured experimentally for fully engulfing fires (Figure 7-23). This same trend was

154

observed numerically. The failure region generated by a local fire impingement is smaller than

that generated by a fully engulfing fire since less material is severely heated.

Figure 7-48: Experimentally observed rupture of 500 US gallon pressure vessel with small

insulation defect (8% of tank surface); Peak wall temp=700oC, peak internal pressure = 2.59MPa (375 psig).

155

Chapter 8 Conclusions and Recommendations

A numerical study was performed to examine the structural failure of pressure vessels

exposed to accidental fires. The MPC Omega method and the Kachanov One-State Variable

technique were both employed to model SA 455 steel’s creep rupture behaviour. Creep rupture is

usually analyzed on the time scale of hundreds or thousands of hours; therefore it was not clear

which creep damage model would best model the creep rupture of SA 455 steel occurring on the

scale of minutes. From the literature review, two techniques were chosen with the hope that

either one or both of them would be able to numerically predict the relatively fast creep ruptures

of pressure vessels. Also, since this analysis involved non-linear FEA, both ANSYS and

ABAQUS were employed to ensure that a solution could be reached. The pressure vessel

analyses modeled 500, 1000 and 33000 US gallon pressure vessels in fully engulfing fire

conditions and a 500 US gallon pressure vessel in various local fire impingement conditions.

8.1 Conclusions The current work developed numerical analyses that modeled the behaviour of pressure

vessels exposed to accidental fires. From the current work, the following conclusions were made.

8.1.1 Creep Damage Methods

• Both the MPC Omega method and the Kachanov One-State Variable technique were

shown to accurately predict the uniaxial creep failure times observed in the creep rupture

experiments.

156

• The MPC Omega method was more accurate then the Kachanov One-State Variable

technique at predicting the uniaxial creep failure strains observed in the creep rupture

experiments.

• The MPC Omega method was more numerically stable then the Kachanov One-State

Variable technique for analyzing the relatively fast creep rupture of SA 455 steel.

• Due to its numerical robustness, the MPC Omega method was used in all the various

pressure vessel numerical analyses while the One-State Variable technique was only used

in some of the analyses.

8.1.2 Commercial Finite Element Analysis Computer Codes

• Both ANSYS and ABAQUS performed adequately when analyzing the uniaxial tensile

specimens to reproduce the experimental creep rupture results.

• When modeling the pressure vessels, ABAQUS clearly outperformed ANSYS.

Numerous convergence issues were encountered with ANSYS, and in the end all the

pressure vessel analyses were conducted with ABAQUS.

8.1.3 Fully Engulfing Fire

• Pressure vessels with a lower length to diameter ratio take longer to fail in a fully

engulfing fire because they benefit from end effects which make it more difficult to bend

a shorter pressure vessel.

• Pressure vessels with a higher thickness to diameter ratio are much safer in a fully

engulfing fire because they have lower wall stresses.

• Pressure vessels with a higher length to diameter ratio are more likely to experience a

BLEVE in a fully engulfing fire since their failure zone was predicted to be larger.

157

Pressure vessels with a lower thickness to diameter ratio are much more likely to

experience a BLEVE in a fully engulfing fire since their failure zone was predicted to be

much larger. These differences in potential BLEVE failure occurrence were predicted

based on larger stresses that occurred in the failure zone due to the different vessel

dimensions.

• A greater fire intensity will generate greater peak wall temperatures and internal

pressures and will produce quicker pressure vessel failures. This is because the vessel

wall material weakens with temperature increase and, hence, will creep faster. Also, the

stresses in the vessel wall will increase with increasing pressure.

• Stress redistribution occurred only briefly at the beginning of the analyses and then stress

levels in the area of peak wall temperatures remained almost constant for the duration of

the analyses. This occurred because the entire vapour wetted region was at high

temperatures and thus there was not much stress redistribution that could occur since load

equilibrium must be maintained. Therefore, a fully engulfing fire makes the pressure

vessel more structurally unstable than a local hot spot.

8.1.4 Local Fire Impingement (Hot Spots)

• The greater the size of the fire impingement, the quicker the pressure vessel will fail.

• Stresses in the area of the local fire impingement can be reduced drastically because there

exists a large amount of cooler pressure vessel wall material that will carry load and

maintain load equilibrium. This means that peak wall temperatures in hot spots can be

larger than those in fully engulfing fire cases and still have longer failure times.

158

• Failure regions generated by local fire impingement are smaller than those generated by a

fully engulfing fire. Therefore, a fully engulfing fire is more likely to cause a pressure

vessel to BLEVE.

8.2 Recommendations

• To further simplify the process of separating high temperature creep and plastic

deformation, creep rupture experiments should be performed with constant true stress

loading instead of constant load or constant engineering stress loading. This would

involve setting up a feedback loop that would decrease the load based on specimen

deflection.

• To improve the accuracy of the creep and creep damage models, a high temperature

extensometer should be used to measure specimen creep deflection in the lab instead of

using the Instron’s linear variable differential transformer (LVDT).

• The mechanical testing could be done with round tensile specimens to minimize the

stress concentration effects of edge imperfections.

• A full transient heat transfer analysis can be coupled with the structural analysis to fully

model all of the occuring heat transfer phenomena as well as the changing fill level of the

pressure vessel. This would be another project on its own to implement into the FEA but

it would make the analysis more complete, as the resulting analysis would not rely on so

many assumptions to be made about the loading conditions of the pressure vessel.

159

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164

Appendix A

SA 455 Steel Measured Creep Curves

165

0

0.05

0.1

0.15

0.2

0.25

0.3

0 200 400 600 800 1000 1200 1400

Time (s)

True

Cre

ep S

trai

n

Experimental 327 MPaExperimental 296 MPaExperimental 266 MPa

Figure A-1: SA 455 steel creep curve; T=550oC.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1000 2000 3000 4000 5000

Time (s)

True

Cre

ep S

trai

n



166

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 500 1000 1500 2000 2500 3000

Time (s)

True

Cre

ep S

train

Experimental 130 MPa




0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 500 1000 1500 2000

Time (s)

True

Cre

ep S

trai

n



167

0

0.1

0.2

0.3

0.4

0.5

0.6

0 500 1000 1500 2000 2500

Time (s)

True

Cre

ep S

trai

n



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1000 2000 3000 4000 5000 6000

Time (s)

True

Cre

ep S

trai

n



168

Appendix B Detailed Drawings of 500, 1,000 and 33,000 US gallon Pressure Vessels

169

CUSTOMERQUEEN'S

DR.MF

CRN REV 2

DATE

SHEET1 OF 1

REV 0

CRN REF. DWG200016

APPROVED BY

DWGPRELIMINARY

DATE27/02/2004

SIZE D

SCALEN.T.S.

500 USWG PROPANE STORAGE TANK FOR QUEEN'S UNIVERSITY

JOB NO.LATER

MANUFACTURING TOLERANCES

.XXX

±.015

±.020

.XXDIMENSION

METAL WORK

±.13

±1°

±.25

ANGLES

UNDER 24"

24" AND UP

FILLET WELD

UNSPECIFIED TOLERANCE

+20%, -0%

DIAMETER WITHIN 1% OF DIAMETER OF VESSEL UNLESS NOTED.

Figure B-1: Detailed engineering drawing of 500 US gallon pressure vessel.

170

Figure B-2: Detailed engineering drawing of 1,000 US gallon pressure vessel.

171

Figure B-3: Detailed engineering drawing of 33,000 US gallon pressure vessel.

172

Appendix C

Results of Numerical Analysis for Time-Independent Temperature and

Pressure Loadings

173

Figure C-1: Predicted MPC Omega creep damage at time of failure, T=620oC p=2.07MPa




174




p=2.24 MPa (325 psig); solved with ABAQUS.

175





176





177





178





179





180

Figure C-15: Predicted effective plastic strain at time of failure, T=680oC p=2.24MPa (325

psig); solved with ABAQUS.

181

Appendix D

Results of Numerical Analysis for Time-Independent Hot Spot Loadings

182

Figure D-1: 500 US Gallon pressure vessel, imposed temperature [oC] distribution for a 100

mm hot spot analysis; maximum temperature of 680oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

Figure D-2: 500 US Gallon pressure vessel, contour plot of MPC Omega creep damage at

time of failure; 100mm radius hot spot, maximum temperature of 680oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

183

Figure D-3: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep

strain at time of failure; 100mm radius hot spot, maximum temperature of 680oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.


strain at time of failure; 100mm radius hot spot, maximum temperature of 680oC, pressure of 2.07MPa (300 psig), solved using ABAQUS; zoomed in view of failure region. Scale same

as D-3.

184

Figure D-5: 500 US Gallon pressure vessel, contour plot of effective plastic strain at time of failure; 100mm radius hot spot, maximum temperature of 680oC, pressure of 2.07MPa (300


Figure D-6: 500 US Gallon pressure vessel, imposed temperature [oC] distribution for a

100mm hot spot analysis; maximum temperature of 700oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

185





186


strain at time of failure; 100mm radius hot spot, maximum temperature of 700oC, pressure of 2.07MPa (300 psig), solved using ABAQUS; zoomed in view of failure region.



187





188





189





190



Figure D-18: 500 US Gallon pressure vessel, contour plot of MPC Omega effective creep strain at time of failure; 75mm radius hot spot, maximum temperature of 720oC, pressure of 2.07MPa (300 psig), solved using ABAQUS.

191





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